MAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS

J. Phys. Chew. Solids Pergamon Press 1959. Vol. 12. pp. 28-40. Printed in Poland.

MAGNETIC PROPERTIES OF POLYCRYSTALLINE
MATERIALS

D. M. GRIMES, R. D. HARRINGTON and A. L. RASMUSSEN

University of Michigan, Ann Arbor, Michigan
National Bureau of Standards, Boulder, Colorado

(Received 9 March 1959)

Abstract--The variation of the magnetic Q with internal magnetization is discussed, using
both the domain rotation and the domain-wall motion model of magnetization change. The var-
iation of the reversible susceptibilities with magnetic moment is reported on four samples, and
the results are compared with results from the frequency spectra in the initial and remanent
states. The distribution of magnetic moments in the system as a function of the angle between
individual and averaged moments is discussed in terms of an infinite series expansion in Legen-
dre polynomials. The coefficients of the first four terms can be measured. Experimental data
are given for the first three.

1. INTRODUCTION net averaged moment of the sample. The rever-
sible susceptibility and resulting differential
THE magnetic susceptibility is defined as the magnetostriction for each mechanism were dis-
ratio of resultant magnetization to magnetic cussed and contrasted in an earlier paper, which
field intensity. In ferrimagnets it is a function will be referred to as I.(l)
of both field strength and frequency. The re-
versible susceptibility is that susceptibility ef- In that paper, the reversible quantities were
fective with a differential magnetic field. The discussed in terms of the distribution of mag-
differential field can be superimposed upon netic moments. The calculations of predicted
a finite biasing field. The reversible suscepti- behavior of the reversible quantities with mag-
bility depends upon the angle between biasing netic moment were carried out with the as-
and differential fields, upon the magnetization sumption that the moments would be distribu-
level in the material, and upon the magnetic ted in some most probable state determined by
history of the specimen. a Boltzmann-type distribution of moments
about the unit sphere. This argument has the
The susceptibility-producing mechanism is merit that the direction of the moments is
not completely understood. It is often assumed strongly influenced by crystallographic orienta-
either that domain moments rotate in unison tion, and that the orientation of neighboring
or that the domain walls move to change the grains can be considered independent. This
does not mean, however, that the resulting mo-
* Supported in part by the U.S. Air Force Office ments must be randomly oriented. A factor of
considerable interest, therefore, is a calculation
of Scientific Research and Development Command of what the distribution of moments is in a
under Contract No. AF 18(603)-S. Reproduction in magnetic material.
whole or in part is permitted for any purpose of the
U.S. Government.

MAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS 29

The detailed calculation off(O) is possible, the anisotropy field remains much larger than
at this time, in only the simplest cases. The the applied field.
basic difficulty is that the solution of this pro-
blem must involve the solution of essentially The reversible susceptibility, as defined
the same number of coupled differential equa- above, is reversible in the thermodynamic sense
tions as there are atomic moments. If the mo- of zero energy dissipation for only the special
ments are treated in the aggregate, localized case of zero applied frequency. An associated
potential minima between neighboring grains energy loss exists for all nonzero frequencies.
and neighboring domains must be included, as This loss is describable as the imaginary part
must the effect of nonmagnetic inclusions. The of the complex magnetic susceptibility, (z =
dominant role played by this type of surface z’ - jz”). The magnetic Q is defined to be
energy has been stressed by GOODENOUGH@). the ratio of real to imaginary susceptibility. It
is convenient in the following section to dis-
The calculation of f(G), then, requires first cuss the loss in terms of Q. The dependence
the development of as realistic as possible a mi- of Q upon ma~eti~tion, field directions, and
croscopic model of magnetic behavior followed source of susceptibility is briefly discussed in
with the distribution of this behavior over the this paper.
sample. The microscopic models used in I
were such that: (A) that the domain moments Frequency spectra have commonly been used
remained always oriented along the crystallo- to investigate the susceptibility sources. This
graphic directions whichminimizedtheanisotrc- extensively used method depends upon the
py energy, and (B) that the static moments rema- different dependence of the frequency of sus-
ined along “easy” directions, but that the sus- ceptibi~~ fall-off on ma~etization mechanism
ceptibility arose by the rotation of the domain to distinguish between sources.(5--8) Some in-
moments away from those directions in the dy- vestigators have also measured the spectra of
namic case. Neither model is completely cor- the material magnetized to remanence and have
rect. Both are highly idealized and, for exam- utilized these results to aid spectral interpreta-
ple, contain neither the “curling” concept dis- tion.@-lr) EPSTEIN and BIRKS(~~) in parti-
cussed by BROWN(~) and by FREI et al.,c4) cular have utilized spectral information to ob-
nor the effect of inhomogeneous fields. A more tain quantitative dete~a~ons of the amount
correct formulation awaits the inclusion of of w~l-motions and domain-rotational sus-
more accurate microscopic models. ceptibility.

There has, however, been ample precedent 2. DISCUSSION
set for considering magnetic susceptibility as
being due either to domain rotation or to do- The macroscopic material is considered to
main-wall motion. This lamentable state is be composed of randomly oriented crystallites.
continued in the present paper. The crystallites are sufficiently small so that
the moment of each crystallite is much less
It was assumed in I that the frequency of the than the aggregate moment of the sample.
measuring field was much less than the reso- The crystalline orientation of each grain is
nance or relaxation frequency of the magnet. taken to be independent of the orientation of
Under these conditions, the magnetization-de- its neighbors. For such a model the static mo-
pendence of the transverse susceptibility on the ment is always parallel with the static field.
biasing field is distinctly different for each of
the assumed susceptibility sources as long as

30 D. M. GRIMES, R. D. HARRINGTON and A. L. RASMUSSEN

The distribution f(O) sin 0 d@ is defined to M-H loop. The coefficient A,/A, can be de-
be the fraction of the atomic moments in the termined from equation (5) and the magneto-
system oriented at an angle between 0 and strictiorP4J5) and independently from equa-
0 + d8 with respect to the applied magnetic tions (9) and (10) of I, where it is stated that
field. It is convenient to expand f(0) in an the susceptibility due to domain rotation is
infinite series of Legendre factions such that: given by:

F(O) = 2 A,P, (cos S) (11 xp = 5 x0(1- (COG0))
n=o I
(6)
Xf = $ x0( 1+ (COS”0,) .
where A, is a function of the magnetic field I
present and the magnetic history of the spec-
imen. AS/A, can be determined from a knowledge
of Al/A, and the use of equations (16) of I,
By definition,
relating the differential ma~etostrictions,
SdfZ(cos” cos"@f(@> namely :
@> = (2)

fdQf@) d, = + d,((cos 0) - tcos3 0,)
d, = + d,(<cos 0, + (co9 0,)
where (cosn 0) represents the average value 1. (7)
of (COS”0) over the sample. Substituting equa- I
tion (1) into equation (2) and integrating over
the unit sphere yields:

o2os”o~= LiA _o m_l j r”dy f’,(r). (3) where $ = F.
2A, s

Upon solving for the coefficients A,,, in equa- (b) Magnetic Q (Domain rotation)
tion (3), using the orthogonality of the Le-
gendre functions, To estimate the magnetization-dependence
of the magnetic Q when the susceptibility
2A = (2m+ 1) (~~(COS~)). (41 arises from domain rotation, it is convenient
to start with a single-crystal fe~oma~et. The
Thus the coefficient of the mth term in the domain-rotation effects are assumed to obey
infinite series expansion for the distribution the differential equatiorW
function can be determined if ( cos O), (cod 0 ),
... (CO@6% are known. So, of course, the first aM H)-KMax aM (81
coefficients are:
,,=dMx dt,

Al/A, = 3 (cos O> (9 where M is the magnetic moment, y is the
AS/A, = 5/2 [3 <co@ 0) - l] ma~etomechanic~ ratio, MS is the spon-
AS/A, = 712 [5 (co9 0) - 3 (cos @>I . I taneous moment, H is the applied magnetic
field, and a is a dimensionless parameter pro-
Thus the distribution function can be experi- portional to the power loss. The total sample

mentally determined if (COP 0) can be meas- moment is assumed to be the sum of moments
ured. The A, coefficients are factions of from all crystallites. It is thus necessary to

both magnetic field and magnetic history. The first calculate intergranular effects, then to
coefficient AJA, can be determined from the sum up all moments over the polyc~st~~ne

sample.

MAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS 31

PARK@) showed that if the magnetic mo- l- 2 $(l- CJ’) (12)
ment of the grains neighboring a given grain ()
averages that of the gross material, then grain
size and particle interaction is described by 1+ o *(G’
a single constant p, defined by the equation ( WO 1

H=h-pM (9) where o is the applied radian frequency, and

where A is the effective differential field, w. = yH,. H,, in turn, is the sum of the static
h is the applied differential field, and M is the applied and anisotropy fields, Hap and Ha,, and
the effective internal field pM.

gross magnetization. The constant p is a fimc- Thus :

tion of the localized packing and remains Ht = Hap+ H,n+pM. (13)
essentially independent of M; p can be re- In the low-frequency limit:

garded as the demagnetizing factor of the grain

partially canceled by the moments of its neigh- QC!!& (14)
bors.

Upon substituting equation (9) into equa- Note that Qr is independent of field direction.
tion (8) and using the additional assumptions The low-frequency initial susceptibility is ob-
that a large static biasing field is oriented in

the Z-direction and sinusoidal time-depen- tained by combining equations (5) and (10) of
dence of an additional applied field, the rever- I. The result is:

sible susceptibility matrix is given by: 20,X0 = (15)
(10)
(equation (8) of I) 3WO
x=f
[1+ tcos” @>I(X_+ X+), 2 ccos 0) (L- X+), 0

- 2 (COS0) (X_- X+), [l+ <cos*@>(Ix-i- x+),0 ,

l0, 0, 2 [l- <cos*O,] (X-C X+> i

where ( ) represents the average value in the where CD~= yM,. Combining equations (14)

polycrystal and 0 represents the angle be- and (15),

tween the spontaneous moment and the appli- xoQr= 201 (16)
ed field. The susceptibilities z, are defined by
3aw.

the equation The product depends upon a and directly mea-

(11) surable quantities, and does not explicitly con-

tain the biasing magnetic field through oo.

where M, and H, are the components of the (c) Magnetic Q (Wd motion)
differential magnetization and field in the The change in magnetic moment due to 180”
x-direction and i = ~“-1. The i operator re-
presents a spatial rotation of n/2 radians. The wall movement is, in unit volume, given by:
resultant algebra from combining equations (8),
(9), and (11) for the Q of the elements on the AM=~2MsA,xk (17)
diagonal of equation (lo), yields:
k

where A, is the area of the Kth wall, and xk is

32 D. M. GRIMES, R. D. BARRINGTON and A. L. RASMUSSEN

the distance through which it is moved by an If the average stiffness term fk increases with
applied magnetic field H. The wall movement
is presumed to be governed by the differential magnetization, then so will Q. Since both A,
equation:
and fk vary with field orientation, the parallel
where fk is the restoring constant for the kth
and transverse Q’s will, in general, differ. This
wall, ,d depends upon the structure-insensitive feature is distinctly different from domain ro-
properties of the material and the parameter tation.
a of equation (8), while m depends only upon
the structure-insensitive properties of the ma- (d) InitiaE and remanent susLgtibility
terial. /co is the permeability of free space. The initial susceptibility due to domain-wall

Combining equations ( 17) and (18), the wall- motion has been approximated by BOZORTH(~~)
motional reversible susceptibility under sinus- for sinusoidal internal strains arising from mag-
oidal excitation is found to be: netostrictive forces present when annealed ma-
terial is cooled below the Curie temperature.
The result is:

4~0 M:
x0 = 3~15E

where 1, is the saturation magnetostriction and
E is the Young’s modulus of the material. Like-
wise the initial susceptibi~~ due to domain

(19) rotation can be approximated as:

In the low-frequency limit, the real suscepti- 2~0M:
bility and the Q become:
x,= 3K,

xw = 4~, M: where p. is the permeability of free space and
K1 is the first-order anisotropy constant. From
Thus for a given material at a fixed low fre- equations (23) and (24) it is apparent that in
pure material the relative importance of the
quency, the Q varies only with ,& A,, and fk. two susceptibility mechanisms depends upon
the relative magnitudes of the effective aniso-
The reversible susceptibility is given as a tropy and magnetostrictive energy densities in
function of magnetization not only by the de- the material.
tailed mech~stic*equation (20) above, but also
by equations (2) and (18) of I. From these, if For material with susceptibility arising by
5 = $’ = II, for virgin material with M = 0 domain rotation, the relationship between the
and if xP decreases montonically with incre- initial susceptibi~~ and the resonant frequency
asing M, then so must xt. Indeed the suscepti- should satisfy the equation:
bilities are related by the equation:
(25)

where y is a constant taken to be 2.21 x IO5

m/A-set, f. is the resonant frequency, MS is the

spontaneous moment of the material, and x0
is the initial susceptibility.

MAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS 33

BECKER and DORING@) discuss a model tation. Their analysis contains the assump-

of magnetic remanence in which, as the field tion that the reversible quantities are due to

is decreased from the saturation value to zero, rotation and that the expansion of magneti-

the moments rotate to occupy the same “easy” zation in terms of applied field must obey the

crystallographic directions occupied in virgin rotational equations through second order in

material, except that all components initially the ratio of applied to anisotropy fields, as

antiparallel to the saturation direction become can be seen from their equation (10). Their

parallel. For this model, the remanent magnet- resulting equation (29) can be put in the form:
ization is 05 M, and the remanent rotational

reversible susceptibilities equal the initial ro-

tational susceptibility. This model is valid for

hexagonal material with K, > 0. FOMENKC(~) for comparison with experimental values.

used it to interpret his permeability spectrum

results. If this is also the position of maximum 3. EXPERIMENTAL

parallel and minimum transverse field sus- (a) Samples measured
ceptibility, then going around the M-H loop
the parallel susceptibility peak occurs for 1M’ Four samples were subjected to detailed mea-

decreasing in magnitude. The remanent and surements and are reported here. The compo-
sitions of the samples are listed in Table 2.
initial susceptibilities will be equal.

The rotational model of gross flux change All except the magnesium ferrite sample were

in cubic crystals would predict moments orien- fabricated at the University of Michigan.

ted in the “easy” directions nearest the field Table 2. Composition of ferrites surveyed
directions at remanence. For this case the re-
manent and rotational susceptibilities taken Designation Composition

from the results of I are given in Table 1 for F-l-2 Ni 0.487Zr,.,s, I% 0,
all anisotropy coefficients save K1 = 0. (These F-6-2 Ni 0’168Z~0.w Coo.zseFc, 0,
are the expected values from reference (1) AA-107-4
for 71-+ CG). I-15-1 Fe, Y, %

Mg Fe, 0,

Table 1. Remanent results MJM, The samples with designation starting F-l
and F-6 were prepared by first ball-milling
~Hexago- Cubic 1Cubic ~Hexago- the C.P. oxides weighed to the desired com-
position in acetone, then decanting, drying,
’ nal / i nal and pressing into a toroidal pill. They were
then heated rapidly to 115O”C, heated slowly
jK,>O’ K,cOIK,>O, K, < 0 to 1375°C for +$ hr, slowly cooled to 12OO”C,
and held for 2 hr. The furnace was then flu-
_ 0500 0.866 0.831 i 0.785 shed with nitrogen and de-energized. The
WM, I-15-1 core was prepared by Dr. D. L. FRESH
x;lx: 1.ooo 0.366 0.449 ' 0.500 and is from the same material as that reported
x:1x: on previously by RADO et aI. as their
1.ooo 1.318 1.276 1.250 type-F core. The AA-107-4 core was prepared
by firing in air at 1350°C for 4 hr. The mixing
I~

An analysis of the relationships between re-
manent condition and the reversible suscepti-
bility has been carried out by FREI and
SHTRIKMAK(~~) on the assumption that the
reversible susceptibility is due to domain ro-

5

34 D. M. GRIMES, R. D. HARRINGTON and A. L. RASMUSSEN

and pressing procedure was the same as for inary susceptibility is seen to be about 130
the type-F-l and F-6 cores. MC for F-6-1, with a corresponding low-fre-
quency susceptibility of 47. For F-6-17, the
(b) Procedures and techniques resonant frequency is about 75 MC, with an
initial low-frequency susceptibility of about
All measurements were taken at ambient 82. The remanent loss for F-6-17 also peaked
temperatures on toroidal samples, since this at 80 MC, with a low-frequency susceptibility
geometry avoids the complexities of demagnet- of 75. An instability of a few per cent was
izing factors. Coaxial line techniques were noted in both samples between different mea-
utilized for the spectral measurements. The suring runs.
complex susceptibility of the F-6 samples was
measured from 1.O to 40 MC, using the radio- 50 IO I00 lml IOOKl
frequency permeameter in conjunction with
a Q-meter. For the F-l samples the real 40 FREOUENCY, MC
susceptibility from 0.1 to 40 MC and the imag-
inary susceptibility from 0.1 to 15 MC were Lx 30 FIG. 1.
measured with the permeameter. A fixed- 0,.
length coaxial line and a radio-frequency a
bridge were used for the remaining higher x 20
loss measurements up to 50 MC. From 50
to 5000 MC, variable-length coaxial-cavity IO
methods were used for complex susceptibility
measurements on both types of samples. 0
I
The susceptibility and Q for the variable-
field measurements were taken with a Q-meter. 100 = INITIAL STATE
For the transverse field measurements, the 80 . ,.: REMANENT STATE
sample was placed between the pole faces of
an electromagnet. The biasing field was along *x 60 IO 100 1000 IOOQO
the axis of the toroid. Girdle windings about 0 FREOUENCY. MC
both outer and inner periphery of the toroid 5
were used to determine the field and magnet- -x 40 FIG. 2.
ization. An extra winding was placed about
the toroid to furnish the biasing magnetic 20
field when the parallel field measurements were
taken. 0
I
(i) F-6 samples. Two different samples of
F-6 material, code designated F-6-l and Comparing the initial and remanent curves,
F-6-17, were measured for the frequency it is apparent that the positions of the peak
spectra. The initial spectrum of F-6-l is shown in both real and imaginary susceptibilities
in Fig. 1. Both the initial and the remanent occur at essentially the same frequency, and
parallel spectra of F-6-17 are shown in Fig. 2. that the ratio of remanent to initial suscepti-
All spectra show but a single resonant peak bility is about 0.91.
in the real susceptibility. The resonant fre-
quency as determined by the peak in the imag-

MAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS 35

The varation of the low-frequency suscepti- tial spectrum of F-l-6 was also measured and
bility with magnetization is depicted in Fig. 3. found to be similar in all respects to F-l-5.
These data were taken on sample F-6-2. The The resonant frequency of F-l-5 was mea-
transverse susceptibility passes through a de- sured at 4.5 MC as determined by the peak
cided minimum near it4 = 0. Both the meas- in the imaginary value. The low-frequency
ured Q’s are essentially constant over a wide susceptibility was measured as 561. The ratio
range of ma~etization, and indeed were never of remanent to initial value of sus~eptibi~~
observed to rise to a value equal to twice the was O-66. The peak in imaginary susceptibi-
initial value. lity at remanence occurred at about 6 MC.

E: The susceptibility was measured as a func-

I8 I2 I20 I2
I6 IO 00 IO
80 08
IO 04 ’ 6-o
08 02 40 04
20 02

-08 -04 0 04 08

MM- M/MS -

FIG. 3.’ Parallel and transverse field susceptibilities; FIG. 5. Parallel and transverse field susceptibilities;
Core F-6-2. Core F-l-2.

(ii) F-l samples. The spectra of the sample tion of the magnetization on specimen F-l-2

code designated F-l-5 were tested. Fig. 4 (see Fig. 5). For this core, the transverse sus-

ceptibility remained essentially constant over

600 a wide range of magnetization values. The
Q’s pass through a minimum near M = 0.

As in the case of F-6-2, the transverse suscep-

tibility remained always larger than the par-

allel value. The Q’s increased rapidly with

increasing field to a value more than 10 times

the initial value.

(iii) 1-15-l. The frequency spectrum for this

material has been published by l?ADO et al.fzo)

0 and corresponds to their “Ferrite F”. They

01 IO IO 100 1000 IOCQO conclude that the low-frequency susceptibi-

FREOUENCY, MC lity is predominately due to the movement of

FIG. 4. domain walls. The variation of the suscep-

tibilities with magnetization is shown in Fig. 6.

shows the complex susceptibility for both For this core, both the susceptibilities pass

initial and parallel remanent states. The ini- through a maximum in the vicinity of zero

5*

36 D. M. GRIMES, R. D. HARRINGTON and A. L. RASMUSSEN

moment, while the Q’s pass through a mini- calculated from equation (25). The experi-
mum. mental value of f,, was determined by the
frequency of the peak value of imaginary sus-
(iv) &l-107-4. The frequency spectra for ceptibility.
this material have been published by two of

Table 3. Resonant frequency (MC).
I2 .______

I Sample type / Measured I Calculated I iE@$$

10

08 F-6-l 179 I
$J- F-6-17
F-l-2 102 0.91
06 + 15 0.66

I 04

I 02 The expected variation of the parallel sus-
ceptibility with magnetization can be com-
FIG. 6. Parallel M/MS - susceptibilities; puted from the transverse susceptibility de-
pendence. For the case of wall-motional sus-
and transverse field ceptibility, equation (22) is the proper equa-
Core I-15-1. tion. For the case of rotational susceptibility,
equation (10) of I is the proper equation. For
us.@‘) It was concluded on the basis of spectral equation (10) to be useful, however, the var-
information that the susceptibility was pre- iation of x0 with applied field must be known.
dominantely due to domain-wall motion. The From equation (15) and the definition of oO,
variation of the susceptibilities with magnet- it follows that in the absence of a detailed
ization is shown in Fig. 7. The transverse susce- knowledge of the effective anisotropy field,
ptibility does not pass through a minimum. which is always the case for polycrystalline
material, x0 is known only so long as the bias-
JO Ik+ ing field is small compared with the aniso-
tropy field.
302 12
2 KJ The magnetic parameters of the four cores
are listed in Table 4.
$2 08 c
t Plots of xP computed from xr for each of the
four samples measured are shown in Figs. 8-11
I 06’ and compared with the experimentally meas-
ured curves. It is apparent that the rotational
I 04 curve fits F-6-2 well and that the wall-mo-
tional curve fits I-15-1 well. The fit is not
0 02 good for either of the other specimens, but
nevertheless the wall-motional curve fits AA-
-08 -04 0 04 08 107 closer than the rotational curve. The con-
MiM< - ._ susceptibilities; clusion that cores F-6-2 and I-15-1 have sus-
ceptibility arising predominantly from rota-
FIG. 7. Parallel and transverse field tion and wall motion respectively is substan-
Core AA-107-4.

1. INTERPRETATION OF RESULTS

The values of f,, measured experimentally

are listed in Table 3, along with the values

MAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS 37

T--a..b-l..e----..4-.._- ..-M-_a_g~n_e_ti_c parameters of four ferrim-_-~-a-g..t.z.-e.tic cores (320--_.k-.c_-)-___-----...-.-
I
Parallel Transverse
._,I _ .--l _“__“.. ___..___________.___, _- -
-_. _--_ -.- ..-
COX -... 1-._z-.,._,-t. -1- Q, j x,, j Q!,_~ 1 %,r /_y___---

F-6-2 /-/
F-1-2*
I-15-1 j 410 ’
AA-107-4 i-/
1 36.4 /

* Values at 500 kc.
t The subscript 0 indicates measurements in the virgin state, p indicates parallel and t trans-
verse measurements. p. and ts represent measurements around the M-H loop at the position M = 0
in parallel and transverse fields. pr and tr represent similar measurements at the remanent position.

12
i-0
00%
$
06

04

02

FIG. 8. Parallel reversible susceptibilities; Core F-6-Z. M/Ms.---e

FIG. 10. Parallel reversible susceptibilities; Core AA-
107-4.

FIG. 9. Parallel reversible susceptibihties; Core F-l-2.

tiated by the spectral interpretation of Figs. 1 FIG. 11. Parallel reversible susceptibilities; Core I-15-1
and 2 and by RADO et al.@@
F-6-2 contains cobalt, and therefore will have
The curve calculated for xP for rotation is a large anisotropy field. Core F-l-2 s a mixed
valid so long as the anisotropy field remains
quch larger than the applied biasing field.

38 D. M. GRIMES, R. D. HARRINGTON and A. L. RASMUSSEN

nickel-zinc ferrite with a corresponding small anisotropy, but does satisfy the conditions of
anisotropy. Thus the lack of agreement with FREI and SHTRIKMAN. Table 5 compares
rotational curves is not significant. For the the values of remanent moment calculated

_._-- ____-. Table 5
’ Measured
Specimen i Calcuiated / H 1
- - _.._~~~~_~_~ .‘_.-_ MrIM__s-.
__ WM, c I x M1_0_S_-_b- --i._. -x.__M1. 0r-S-.-_

AA-107-4 0.389 , 0.259 160 1,06 1 0.41
F-6-2 127 3.58 ! 237
F-I-2 0.660 / 0.041 100 / 3.67
2.24
0.610 o-330

I-15-1 0.730 0.675 175 ’ 1.35 0.98
I._ __-_ ..~I_ _. m_==_z._ -I__ ._..._. _

F-l-2 core, there was a slight but real mini- from equation (26) with experimentally meas-
mum in the xr in the vinicity of zero moment.
This feature is not consistent with wall mo- ured values. The agreement is not good, and
tion and the accompanying ?I,,behavior. There- indeed the best agreement is with the material
fore it is concluded that domain rotation whose susceptibility arises from wall motion.
must be present. No such behavior was found
in AA-107. In the spectral data, two peaks in Upon equating x0 from equation (6>, the
the imaginary susceptibility were observed average value of tcos2 0) is found to be:
for AA-107, and only one for F-l-2. There
must, therefore, be two impo~ant mech~sms Solving for tcos”@ from equations (7),
for the AA-107. These mechanisms are assum-
ed to be rotation and wall motion. For F-l-2, (cos%@ = (is) ccos@>. (28)
one can say only that domain-rotational effects
enter. If w~l-mo~onal effects are present, the Note that (cos’O> should vary from l/3 in the
resonant frequency must occur at the same demagnetized state to unity when the material
point as the rotational effect. The essential is saturated. A similar we-motions equation
difference between F-l and F-6 lies in the yields :
value of the anisotropy. Since rotational effects
have been shown to exist in the high-aniso- ( )1__2 dlnxf
tropy material, it is expected that they should &t-- XP dlnA4
also exist in the low-anisotropy material.
2xtf xp = -_3_--__q-.g-34 * (2g)
Material F-6-2 apparently fits the condi-
tions imposed in this paper for the static con- If the values of xr and M are calculated assum-
ditions, nameIy oriented along easy crystallo-
graphic directions, and the susceptibility obeys ing a Boltzmann ~stribution of moments,
the rotational equations. This material would then x1 = Ai%f,L(?$l~ and M = i&L(q), where
not satisfy the conditions of FREI and A is a constant for the material, L(9) represents
SHTRIKMAN@~). On the other hand, F-l-2 the Langevin function of q., and ~1= AH,,
does not satisfy the conditions as described
in this paper, apparently because of the small

MAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS 39

where Ht is the totalized biasing field, includ- to domain rotation can be interpreted in terms
ing history, anisotropy, and applied field. of variation from the most probable condition.
Substituting these values into equation (29) For F-6-2, as the moment is decreased, mo-
yields a function which goes from l/3 for M = ments parallel and antiparallel are initially in
0 to unity for saturation. excess of the most probable condition. How-
ever, as M is further decreased towards zero, the
Plots of *XL as a function of (cos 0) are parallel-antiparallel components increase to lar-
ger than the most probable value. This can be
Xt P considered the reason for the transverse sus-
ceptibility at zero moments being larger than
shown in Figs. 12 and 13. In Fig. 12, the the- the initial susceptibility, and the corresponding
oretical curve is the value of tcos” 0) calculated parallel susceptibility at zero moment being
for domain rotation. In Fig. 13, the theoretical smaller than either. Sample F-l-2, if the sus-
ceptibility is assumed entirely rotation, main-
06 tains at all times a predominant parallel-anti-
05 parallel moment configuration. The value of A,
04 from equation (5) as a function of magnetic mo-
03 ment can be read directly from Fig. 12 for
sample F-6-2.
-00 -04 0 04 06
M/MS - More detailed determinations and analyses
FIG. 12. Variation with (COS8). of the distribution functions in other materials
of (cos’ 0) must await simultaneous parallel and transverse
susceptibility and differential magnetostriction
measurements.

5. CONCLUSIONS

The magnetic properties of a ferromagnetic
system can be described in terms of a distribu-

tion of magnetic moments about the unit sphe-

06

re. This function cannot be calculated in detail

for macroscopic systems. However, assuming
idealized models of magnetic behavior such as

magnetization by domain-wall motion or mag-
netization by domain rotation, some reversible

-08 -09 1 \? d? 1 1 properties of ferromagnets can be compared

FIG. 13. --- 0 04 00 without detailed knowledge of the distribution
c::, I function. If a detailed distribution function is
M/MS- assumed, then, of course, the magnetic proper-
ties can be calculated in detail.
::Iversus M/M,.

I

curve is based upon domain-wall motion. Both The inverse of this calculation can also be
curves are based upon Boltzmann distribution made. The procedure consists of firstly expand-
of moments. The rotational curve attributed ing the distribution function in an infinite ser-

40 D. M. GRIMES, R. D. HARRINGTON and A. L. RASMUSSEN

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