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Magnetic properties in textured fine-particles magnets
Journal of Magnetism and Magnetic Materials 43 (1984) 317-326 North-Holland, Amsterdam
317
MAGNETIC PROPERTIES IN T E X T U R E D F I N E - P A R T I C L E S M A G N E T S B. S Z P U N A R and J. S Z P U N A R Physics Department, University of Durham, Durham, UK
Received 25 May 1983; in revised form 3 February 1984
The effect of texture on the magnetic properties of single domain particle magnets is discussed. Analytical formulae are derived which enable us to correlate the initial susceptibility, coercive force, anisotropy field and remanence with series expansion coefficients of Legendre Polynomials describing the texture. The analysis of texture influence on the magnetic properties of alnico 5 is presented. This analysis indicates that the initial susceptibility is most strongly affected by the texture changes. The remanence does not change significantly with the texture. The influence of various texture harmonics on magnetic properties is discussed. Also it is demonstrated that the popular use of simple parameters characterizing texture like mean deviation angle or maximum deviation angle may give incorrect results. The problem of calculating the texture from the magnetic data is discussed.
1. Introduction The effective magnetic anisotropy in polycrystalline fine particle magnets arises not only from the magnetocrystalline anisotropy of the particles or from their shape but is strongly influenced by texture. The meaning of the word texture as used in this paper implies two pieces of information. Firstly the distribution of crystallographic directions and secondly the distribution of the particle orientation. Other forms of anisotropy like stress anisotropy surface or exchange anisotropy will be neglected. If the easy magnetic direction of a particle is determined by the magnetocrystalline anisotropy, information about the texture can be obtained f r o m X-ray or n e u t r o n diffraction experiments. If, however, the magnetic anisotropy of the particle is decided by its shape microscopical observations" provide the experimental data about the texture. These two distribution functions, different in their definition, m a y often be identical, for example an alnico magnet in a finely dispersed state consists of elongated ferromagnetic precipitations in a non-magnetic matrix. The crystallographic texture of this matrix determines the distribution of the easy magnetic axis.
There are examples of important materials where the texture is a decisive factor in obtaining the required magnetic properties; as such we can mention the alnico permanent magnets, S m C o 5 or BaFe12019. Also the orientation of magnetic particles is a continuing problem in the development of superior recording films. The mangetization processes in these magnets can be described assuming that each individual particle or grain constitutes a quasi-single domain. The easy axis of individual particles can be determined b y the magnets crystalline anisotropy or by the particle shape. Usually one type of anisotr o p y dominates. Magnetocrystalline anisotropy is for example, the p r e d o m i n a n t factor in C o - C r recording films [1] or in S m C o 5 magnets while the particle anisotropy shape dominates in most of the alnico alloys. In spite of its importance the influence of the texture on magnetic properties of single d o m a i n magnets has not been studied extensively. McCurrie [2] suggested that the alignment of the easy axis can be derived from remanence measurements. Higuchi and M i y a m o t o [3] c o m p a r e d the mean deviation angle in misorientation of crystals in alnico 8 with the calculated magnetic properties. M o o n [4] c o m p a r e d the measured mean
0 3 0 4 - 8 8 5 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
318
B. Szpunar et al. / Textured fine-particle magnets
deviation angle in alnico 5 specimens with the measured remanence and the coercive force. All these authors used the common assumption that the crystal orientation distribution is constant in the angular interval up to the mean deviation angle. Such an assumption does not agree with texture experiments and as we demonstrate later may lead to an error in the calculation of the initial susceptibility, coercive force and remanence. There is thus a necessity for more precise quantitative information about the texture. Such a quantitative description of the texture [5] is extensively used in order to explain the anisotropy of plastic and elastic properties of polycrystalline metals. Also the magnetic properties like magnetocrystalline energy [6] or magnetostriction [7] in textured materials can be calculated using the series expansion coefficients of the crystal orientation distribution function (ODF). A general description of texture is rather complicated, however, in many practical problems the symmetry of the orientation distribution and the magnetic symmetry of the particle simplifies the description. For the experimental problem discussed in this paper the texture may be described by the series expansion of Legendre's polynomials since in the majority of technical magnets, the orientation of the magnetic particles is symmetrical about the direction of the external magnetic field. The calculation will be carried out by assuming that the particle is a single domain at zero field and it will stay saturated at any field. The same mechanism of rotation of the spin in unison has been used by Higuchi and Miyamoto [3] giving agreement with the measurement of the coercivity. However, other mechanisms, particularly curling, may contribute to the results. Also the assumption will be made that the magnetic particles are dispersed, thus the interaction between them can be neglected.
2. Model
We dealt with an assembly of non-interacting, single domain particles. We assume that the particles' shape is ellipsoidal with the polar axis " O "
and equitorial axis " T " . Also they may have uniaxial magnetocrystalline anisotropy with an easy axis parallel to the " O " direction. For particles having a spherical shape the discussion will be limited to the effect of texture only. The particle magnetic anisotropy is considered to be the same which means here a constant composition or shape for all particles. It is usually the case that the composition is constant. Systems with the same shape are more difficult to find and one has to be satisfied with information about the mean value of the particle shape anisotropy. The assumption of the same magnetic anisotropy of the particle does not mean that the approach cannot be used to calculate the magnetic properties in systems having a variety of particle shapes or even different composition. Since the particles are non-interacting the rule of superposition applies and the magnetic properties can be calculated accordingly. If the particle shape is eUipsoidal and the easy axis does not coincide with the " O " axis, separate information on the distribution of the easy axis is necessary. This case seldom occurs and is not discussed in this paper. Usually one of these two anisotropies (texture or distribution of orientation of ellipsoidal particles) is dominant. The effect of it on the magnetic properties may be calculated by the proposed model. We introduce now two coordinate systems, one attached to the specimen (with three orthogonal axes, X, Y, Z ) and another, that of the particle (axes coincide with " O " and " T " directions). The direction of the magnetic field is defined by the polar angles q~, o in the frame of the specimen and by the angles a, fl in the reference frame of the particle. The orientation distribution function of the magnetic field vector, g+o(a, fl), is defined as the probability that H(qJ, o) coincides with the direction t~, fl in every particle. We take the magnetization to be in the direction M(0, @) (see fig. 1) and the external magnetic field to be in the direction H ( a , fl). The total energy is expressed in the specimen reference frame as a sum of the energies of the magnetocrystalline anisotropy E A, demagnetization energy E D and the energy Eia of the magnetic interaction between the magnetization and the
B. Szpunar et al. / Textured fine-particle magnets
z[
319
energy becomes:
I"I (~ o') H'(ct ~)
fly = ½(K sin2a + K sin2(90 - a ) ) =
K/2.
Fig. 1. Symbols.
This gives the wrong conclusion because if the average anisotropy is not dependent on a the m e a n magnetization can be directed in any direction. We should proceed with the minimization first which gives a m i n i m u m at a = 0 ° for the first particle and at 90 ° for the second particle. The calculated direction of the average magnetization is thus at a = 45 °.
field.
3. Initial susceptibility
~' (0, ¢)
t,,X
E = EA + E D + E H.
(1)
The magnetocrystalline anisotropy energy is given by EA =
K sin20
=
K - K cos20.
(2)
The demagnetization energy can be expressed as: E D = ½M2(N0 -
Nr)cosZO,
(3)
to simplify t h e presentation we introduce
F o r low magnetic field and uniaxial anisotropy minimization of the energy over 0 and q~ can be done analytically. Following Stoner and Wohlfarth, ref. [8] concerning an assembly of particles having shape anisotropy an analytical expression for the initial susceptibility can be derived. Minimization of the energy over ~ can be done for any value of the external magnetic field. The derivation of the energy with respect to ~ gives:
OE/Oq~=
7= ½M2(NT- No) + K
=
and then EA + E D =
- T cos20.
(4)
We should notice that in the expression of the energy the constant terms can always be omitted. The energy of an interaction with the external magnetic field H is expressed by: E H = -MH(sin
et cos/3 sin 0 cos q~
+ sin a sin/3 sin 0 sin ~ + cos 0 cos et).
O E H / O dp
-HMsinasinOsin(fl-qJ)=O,
(6)
which leads to the condition/3 -- ,/,. This result is crucial in obtaining a non-zero projection of the magnetization in the direction of the external magnetic field for a r a n d o m distribution function. Utilizing this condition one obtains the formula for the total energy. E = - T cos20 -
MH c o s ( a - 0).
(7)
Minimizing the energy we get (5)
W h e n attempting to find the formula for the average energy of textured particles we have to stress however, that the procedures of averaging and minimization do not commute. We m a y get wrong result if the E n is averaged over the orientation distribution function first. Consider for example two spherical particles, at a very low magnetic field. Both have uniaxial anisotropy, their easy directions being perpendicular to each other. The formula for the average
OE/OO= - 2 T cos 0 sin 0 - MH s i n ( a - 0) = 0.
(8) F o r the low values of the field strength, which means that IMH/2TI << 1, the magnetization vector is nearly parallel to the Z-axis of the specimen reference frame. Also, as sin 0 << cos 0, it follows therefore that sin 0 = M H ( s i n
a)/2T.
(9)
The projection of the magnetization on the exter-
B. Szpunar et al. / Textured fine-particle magnets
320
hal magnetic field is
M H = M ~ cos(a - O)g¢o(ct , fl)sin a d a dfl. (10) By use of eq. (9) we can proceed now with the calculation of the average magnetization. One finally obtains
Mn=
M 2 H f~ f 2 ~ . 3 2T Jo Jo sinagg, o(a, fl) d a d f l +
~o ~o
1
8T 2
x sin 2a g~o(a, fl) d a dfl.
(lOa)
Because, the orientation distribution function is of an even power with respect to cos a the second part of eq. (lOa) is equal to zero and the following formula for the initial susceptibility is obtained OM~
x
aH._o
M 2
/'~ /'2~r
. 3
results as those of Stoner and Wohlfarth [1]. The derived expression of the susceptibility is also valid for particles having cubic magnetocrystalline anisotropy since for low values of 0 the cubic anisotropy and uniaxial anisotropy are expressed by a similar formula. The use of formula (11) to calculate the susceptibility requires the quantitative representation of the texture. All magnetic parameters are dependent only on the angle a which means that the texture function g~o(a, fl) may be simplified. The general form of the texture function is
"
oo
M(I)
g~o( a, fl) = E
E t'~k~'(a, ill,
(12)
I~0 m=l
where surface spherical harmonics k T ( a , fl) are expressed using associated Legendre functions P~(cos a), namely
TTJoJo slnag¢o(a, fl)dadfl. (11)
For the random distribution (a, fl) = 1 / 4 ~ one obtains
function
g+o
X = ~M2/T. If additionally the magnetocrystalline anisotropy can be neglected ( K = O) one obtains the same
k'/'(a, fl) - ~~
e " ~ -Pt" " (cos ,~).
(13)
Since the texture used for magnetic calculations is only an a function, the measured texture g~o(a, fl) can be integrated over fl or an experiment can be performed which gives automatically the required integration. The texture will thus be described by the ex-
Table t C o n s t a n t s u s e d in the c a l c u l a t i o n o f the i n f l u e n c e o f the t e x t u r e o n the m a g n e t i c p r o p e r t i e s Legendre polynomials
Texturesusceptibility
Textureremanence
Texturecoercivity
Textureanisotropy
Pt(a)
coefficient
coefficient
coefficient
field
Xl
vl
8ct
~Ai
P0 ( a )
0.943
0.707
0.530
1.672
P2 ( a )
- 0.422
0.395
0.215
- 0.145
P4 ( a )
0
- 0.088
- 0.082
0.223
Pr(a)
0
0.040
0.077
- 0.061
Ps ( a )
0
- 0.023
- 0.011
0.032
Plo(a)
0
0.015
0.020
- 0.035
P12(a)
0
-- 0 . 0 1 0
- 0.005
0.051
Pt4 ( a )
0
0.008
0.013
-- 0.023
Pa 6 ( a )
0
- 0.006
- 0.002
0.033
Pls(a)
0
0.005
0.007
- 0.017
P20 ( a )
0
- 0.004
-- 0.002
0.024
B. Szpunar et aL / Textured fine-particle magnets
pression L
g~o(a) = E t , P t ( a ) -
(14)
/=0
From (11) and (14) we get M
X=-~
2
L
F_, I=0
ttf
,n
sin 3a fit(a) d a .
321
is in the direction closest to the magnetic field gives us maximum and minimum values of the remanence. M R for particles having uniaxial anisotropy is theoretically between 0 and 1. For cubic spherical particles having easy direction [100] or [111] the changes of the remanence due to the texture are limited to the interval
(15)
1/7c3
*'0
Introducing a texture-susceptibility coefficient )kt = ~ s i n 2 a .P/(a) d a ,
(16)
<
MR/M s
<
1.
(20)
In materials which have [110] easy direction this interval is smaller, namely
V/~ < M R / M s < 1.
(21)
we obtain M2 ±
X = ~-~ )'~ ttX,.
(17)
I=0
Assuming that the texture function is given by formula (14) we can modify the expression (19) and obtain: L
The values of Xt are given in table 1. Substituting the calculated value for P2(a) one obtains
MR = E tt`/t,
M2 X = - ~ - (0.943to = 0-422t2 ).
where 39 is given by
(18)
where t 0, t 2 are the texture expansion coefficients (t o = 1/v~-). All other integrals over higher order polynomials are zero.
Ms
(22)
t=0 ,ff
"// = ½fo P t ( a ) s i n 2 a d a .
(23)
Calculated values of "/t are given in table 1.
4. Remanence 5. Coercivity Frei et al. [9] suggested the formula for the relationship between the remanence and the distribution of the easy axes of uniaxial ferromagnetics for an assembly of single domain particles. Bunge [5] also used the texture formalism and derived the expression for the remanence in textured hexagonal metals. A similar approach will be used here in order to obtain the coefficients which m a y be used directly to calculate the remanence in textured single domain magnets. The mean relative component of magnetization in the direction of the original field can be calculated from the following formula:
MR:f2. Ms - - ' o "o cos a g,~o(a, fl)sin a d a
We report here Stoner's and Wohlfarth's [8] calculation including additionally the anisotropy described by the orientation distribution function. For the coercive field the following conditions are valid: M c o s ( a - 0) = 0,
0 e / 0 o = o, ~.2E/002 > O.
(25)
These conditions are fulfilled only when a > ~r/4. In this case the coercivity is given by:
H c = ( 2 T / M ) s i n a cos a, dfl,
(26)
(19)
where M R is the remanence and M s is the saturation magnetization. The condition that the domain
and for textured materials this becomes,
•/c=
2T : ' e2~r
-joJ0
• 2
g¢o(a, f l ) s m a c o s a d a d f l .
(27)
322
B. Szpunar et al. / Textured fine-particle magnets
One should stress, here however, that the averaging of the coercivity over the texture function is only an approximation [10], because the assumption of homogeneous magnetization is not fully justified. For a < ~ / 4 the coercivity is equal to the multivalue critical field H M. The conditions for the anisotropy field are in this case O E / O 0 = 0 and OZE/O02 = 0.
(28)
Following Stoner's and Wohlfarth's calculations we have: 2T HM =--~-V/1 -- tz + t4 / ( t2 + 1 ) ,
(29)
3 where t = ~ t g a . For textured materials one obtains 2T fo~fo2 ~rv/1 - t2
+
t4 /(
+1)
× g ¢ o ( a , fl)sin a d a dfl.
(30)
According to eqs. (14), (30) and (4) the coercivity for a symmetrical orientation distribution will be given by
/4c
2T
(31)
tirol, I=O
where the texture coercivity coefficients are defined by |
2] U2 t( )sin2 [~,~/4
+ f0~/4~/1
-
cos do
t 2 + t 4 / ( t 2 + 1)
× P/( )sin
(32)
The anisotropy field H A = 2 H M is given by
4s
HA = 2HM = M
tt~a"
(33)
/=o where 8Ai equals 8A, = f0~V/1 -- t 2 + t 4 / ( t 2 + 1) PI (or)sin et dc~. (34) The texture-coercivity constant 8c~ and the anisot-
ropy constant 8Al are given in table 1 for the different values of I.
6. Discussion
The magnetization in a powder magnet subject to a uniform external field varies from particle to particle, also the internal field on each particle is different due to its own magnetization and the magnetization of neighbouring particles. Since the assumption is made that particles do not interact, the internal field is considered uniform through the magnet and the calculation of the mean magnetization is justified. This simplified assumption is used for calculation of the mean value of the initial susceptibility and the remanence. The calculation of the mean coercivity has been made by averaging the coercivity over the texture function. Such method is justified only if the magnetization throughout the magnet is homogeneous. It has been used previously by other authors, i.e. Lawton and Stewart [11]. The average value of the coercive field obtained using this method for a random texture is 0.375. The result does not agree with the value H c = 0.479 obtained by Stoner and Wohlfarth for a random assembly of particles and a mean value of the magnetization. These two approaches to the calculation of the coercive force are compared in fig. 2 with the experimental results obtained by Paine, Mendelsohn and Luborsky [12]. Our approach is closer to the experimental data obtained for single domain iron particles. Both results represent the two extreme approaches to the problem of averaging the magnetic properties and neither the assumption of zero field from the neighbouring particles nor that of uniform magnetization is fully justified. The calculation of the susceptibility, the coercivity and the remanence in single domain textured magnets requires the knowledge of series expansion coefficients characterizing the texture. The number of coefficients which are necessary is influenced by the orientation distribution function characterizing the examined specimen but is more
323
B. Szpunar et al. / Textured fine-particle magnets I
I
I
I
I
I
t~000
II~
/ ./t
_
,/
f ~LCALCULATEDBY sTONERAND
,~
3000
/ /
uJ"
/
/
,92000_
--
WOHLFARTH~ f
/
/// /
-
PRESENT METHOD
oBsj._~ +
I
I 2
1
I
I 3
L
I t~
1
MEDIAN LENGTH TO DIAMETER RATIO Fig. 2. Relation between medium elongation and coercive force
by Paine, Mendelsohn and Luborsky [12] compared with our calculation.
strongly d e p e n d e n t o n the magnetic property which is calculated. A n a l y s i n g the result given i n table 1 we see that for the calculation of the initial susceptibility only two texture coefficients are required. The texturesusceptibility c o n s t a n t s for l > 4 are all zero. T h e "tz c o n s t a n t s characterizing the value of the reman e n c e decrease with a n increase in l a n d o n l y the
first few texture coefficients are needed. The function describing the coercive field H c a n d the anisotropy field H A is more complex, a n d therefore a higher n u m b e r of associated Legendre p o l y n o m i a l s is required for the calculation of the a n i s o t r o p y field in textured materials. The relationship between the texture a n d the a n i s o t r o p y of m a g n e t i c properties is illustrated b y c o l u m n a r alnico magnets. Texture data are taken from the p a p e r b y M o o n [4]. Histograms illustrating the dispersion of [001] direction a b o u t the m a g n e t axis for the 8 different specimens are analysed a n d the series coefficients characterizing the texture i n these specimens are calculated. Three specimens have their o r i e n t a t i o n distribution f u n c t i o n s in the interval 0 ° - 1 5 ° . F o r the r e m a i n i n g specimens the d i s t r i b u t i o n is slightly broader, however, o n l y negligible intensity is observed at angles higher than 20 ° . F o r the eight analysed o r i e n t a t i o n d i s t r i b u t i o n s the changes in the t 2 texture e x p a n s i o n coefficient is a b o u t 5% of its m e a n value. The higher order coefficients c h a n g e d more strongly i.e. t 4 changes b y a b o u t 20% 16 a b o u t 40% a n d t 8 more t h a n 80%. T h e calculations showed (see e.g. table 2) that the texture most significantly influences the initial susceptibility; the variation in a m p l i t u d e of the coefficients characterizing the texture of the vari-
Table 2 Magnetic parameters characterizing specimens of alnico 5. Remanence and coercivity as measured by Moon [4]. The magnetic parameters as calculated in this paper using the series expansion method for l = 8, 1= 20 and directly for the measured histogram characterizing texture (exp) Specimen
1A
1B
2A
5
Series
Calculated
expansion
susceptibility
remanence
coercivity
anis. f i e l d
Measured
l
x/(M2/2Ke)
dr/Js
H¢/(2Ke/M)
HA/(2K=/M )
8 20 exp 8 20 exp 8 20 exp 8 20 exp
0.040 0.040 0.040 0.028 0.028 0.028 0.018 0.018 0.018 0.016 0.016 0.016
0.969 0.979 0.979 0.972 0.985 0.986 0.973 0.990 0.991 0.973 0.996 0.992
0.665 0.693 0.696 0.676 0.708 0.707 0.686 0.734 0.733 0.688 0.742 0.747
1.380 1.388 1.385 1.414 1.415 1.414 1.452 1.470 1.467 1.460 1.490 1.494
remanence (T)
coercivefield Hc (kA/m)
1.34
55.7
1.34
57.3
1.35
61.7
1.34
59.7
324
B. S z p u n a r et al. / Textured f i n e - p a r t i c l e magnets
ous specimens may change the initial susceptibility by a factor of three. Unfortunately we do not have the experimental data for the susceptibility of the analysed specimens. However, similar changes in the initial susceptibility for alnico alloys have been measured by Higuchi and Miyamoto who demonstrated that the susceptibility changes from 0.024 to 0.092 G / O e (table 3) correspond to a change of the mean deviation angle from 22 ° to 45 ° . The same authors have estimated the deviation angle from the measured susceptibility for two specimens which have the same value of remanence. From the estimated deviation angle they calculated the remanence and as we see in table 3 (specimens 9 and 10) they obtained results which completely disagree with the experimental values of the remanence. This example illustrates the fact that the use of the average deviation angle for characterizing the texture is unsatisfactory. Since susceptibility-texture constants "/i are expressed in terms of sin2et any change in the texture at low angles may produce significant changes in the initial susceptibility. Such changes frequently occur (see fig. 3) but cannot be adequately described using the mean deviation angle. Description of the shape of the texture function using series expansion coefficients offers the possibility of precise calculations. It has already been demonstrated that the initial susceptibility can be calculated knowing only one texture coefficient t 2 since the "//for l > 2 are zero (table 1). For the investigated specimens we observe significant changes in the amplitude of higher order expansion coefficients. Such changes may only affect other magnetic properties such as the remanence, the coercive field and the anisotropy field. The values of the remanence calculated for
20 18 16 1/*
-t=50 --t:36 ....... t=20
12 10 ~:
8
~
----t:8
/
.....
=
2
6 /, 2 0
50
100
150
200
250
Fig. 3. Histogram of texture for 1A specimen. Functions illustrating the series expansion approximation for / = 2, 8, 20, 36 and 50 are added.
analysed specimens do not change significantly (less than 1%). As we see from the definition of the "// coefficients (eq. 23) changes of cos et in the interval (0°-15 °) are small and thus coercivity is not sensitive to changes of orientation density in this interval. It is interesting to note that McCurrie [2] suggested that the extent of the easy axis alignment in uniaxial permanent magnets can be calculated from remanence measurements. We believe that in most technical magnetic materials which exhibit small differences in texture it will be difficult and inaccurate, even if measurements are made in the direction perpendicular to the axis of alignment. As an example (see table 2) the remanence measured in the specimens 1A, 1B and 5 is the same, our calculation also agrees with these experimental results, howex/er the initial susceptibility
Table 3 Results obtained by" Higuchi and Miyamoto [3] Specimen no.
9 10
Observed magnetic properties
Calculated magnetic properties
Deviation angle
Br
H~
X
Br
Hc
Om
(kG)
(kOe)
(G/Oe)
(kG)
(kOe)
(deg)
11.6 11.6
1.52 1.46
0.024 0.066
11.6 10.2
1.68 1.47
2215 37 26
325
B. Szpunar et al. / Textured fine-particle magnets
varies from 0.16 to 0.40. The formula suggested in ref. [2] was derived assuming that there is a constant density of the distribution of the easy axis up to the maximum misorientation angle. As we demonstrated, such an assumption is oversimplified and we suggest the solution which makes use of the experimentally measured orientation distribution. Our results demonstrate that initial susceptibility is most sensitive to texture changes in technical alnico magnets. However its use to determine the texture quantitatively is limited, since it is only sensitive to changes in the t 2 texture coefficient. The same applies to the coercive force, because several texture series expansion coefficients have to be known in order to describe the crystal orientation distribution. The coercive force in the investigated specimens is dependent on the shape of the crystal orientation distribution. The changes of texture in the analysed specimens are, according to the theory, responsible for a 5% variation in the coercive force. Fig. 4 illustrates the relation between the experimental value of the coercive force and the amplitud.e of the texture series expansion coefficients. Contrary to its effect on the initial susceptibility an increase in the amplitude of the second order texture coefficient causes an increase of the coercive force. The changes in the amplitude of the higher order series expansion coefficients contribute significantly to the theoretical value obtained for the coercive force. The contribution of observed changes in the value of t 2 is roughly the same as the contribution of changes in the t 4 coefficient. Changes in the t 6 texture coefficient contribute the most to the observed differences in the coercive field of the analysed specimens. The calculated values for the coercivity (see table 2) are however dependent on higher order texture coefficients. A series expansion up to l = 14 is, for the analysed specimens, sufficient. Termination of the series expansion at l = 14 may however not be sufficient for describing texture, especially in the case of a strong texture where the orientations of crystals are concentrated in a small angular interval. As an illustration (fig. 3) we compare the shape of the function termination at the different values of 1, l = 8, 20, 36 and 50. The series expansion
/
60
t (el t (+) f6(o} t8 (~x)
"6
<:
~ / fi
f .,*
/
59
/
58
/ /
i
I
I
I
t
I
I
12
1~
16
1-8
20,
22
AMPLITUDE OF TEXTURE SERIES EXFANSIONCOEFFIEIENT$
Fig. 4. E x p e r i m e n t a l c o e r c i v e f o r c e as a f u n c t i o n o f t h e a m p l i t u d e o f t h e t e x t u r e series e x p a n s i o n c o e f f i c i e n t s f o r a l n i c o 5.
l = 50 describes the main features of the experimental distribution, however its maximum is lower than that of the experimental histogram.
7. Conclusions Describing the initial susceptibility, the remanence, the coercive field and the anisotropy field using the same orthogonal functions which are used in the description of the texture makes it possible to discuss and to analyse the influence of the texture of the magnetic properties. The formulae for the calculations of these properties in single domain fine particle magnets have been suggested. The coefficients ~1, 71, 8ct, 8 A I have been introduced which characterize the influence of the texture on the various magnetic properties. Accordingly the initial susceptibility is dependent only on one texture coefficient t 2. To calculate other magnetic properties we need however more precise information about the texture and the texture coefficients up to ! = 14 may be needed in the case of alnico type alloys. The remanence in these alloys is not strongly dependent on the observed texture variation. Initial susceptibility however is very sensitive to texture changes. The observed changes in the coercive force with the texture are small, 5%. The quantitative description of texture which
326 has b e e n i n t r o d u c e d has a n descriptions. For example d e v i a t i o n a n g l e m a y l e a d to the initial s u s c e p t i b i l i t y a n d
B. Szpunar et al. / Textured fine-particle magnets
advantage over other the use of the m e a n a n i n c o r r e c t v a l u e for remanence.
Acknowledgements T h e a u t h o r s w i s h to t h a n k D r . B.K. T a n n e r for v a l u a b l e d i s c u s s i o n s a n d are i n d e b t e d to Dr. W . D . C o r n e r for c o m m e n t s o n t h e m a n u s c r i p t . W e g r a t e f u l l y a c k n o w l e d g e the c a r e w i t h w h i c h Mrs. V. T o d d a n d M r s . M . B r a d l e y t y p e d the manuscript.
References [1] s. lwasaki and K. Ouchi, IEEE Trans. Magn. MAG-14 (1978) 849.
[2] R.A. McCurrie, J. Appl. Phys. 52 (9181) 7344. [3] A. Higuchi and T. Miyamoto, IEEE Trans. Magn. MAG-6 (1970) 218. [4] J.R. Moon, J. Phys. D7 (1974) 1233. [5] H.J. Bunge, Texture Analysis in Materials Science (Butterworths, London, 1982). [6] J. Szpunar and M. Ojanen, Met. Trans. 6A (1975) 561. [7] J. Szpunar, J. Appl. Phys. 55 (1984) 2133. [8] E.C. Stoner and E.P. Wohlfarth, Phil. Trans. Roy. Soc. A240 (1948) 599. [9] E.M. Frel, S. Shtrikman and D. Treves, J. Appl. Phys. 36 (1953) 443. [10] E.P. Wohlfarth, private communication. [11] H. Lawton and K.M. Stewart, Proc. Phys. Soc. 63A (1950) 848. [12] T.O. Paine, L.I. Mendelsohn and F.E. Luborsky, Phys. Rev. (1955) 1055.
317
MAGNETIC PROPERTIES IN T E X T U R E D F I N E - P A R T I C L E S M A G N E T S B. S Z P U N A R and J. S Z P U N A R Physics Department, University of Durham, Durham, UK
Received 25 May 1983; in revised form 3 February 1984
The effect of texture on the magnetic properties of single domain particle magnets is discussed. Analytical formulae are derived which enable us to correlate the initial susceptibility, coercive force, anisotropy field and remanence with series expansion coefficients of Legendre Polynomials describing the texture. The analysis of texture influence on the magnetic properties of alnico 5 is presented. This analysis indicates that the initial susceptibility is most strongly affected by the texture changes. The remanence does not change significantly with the texture. The influence of various texture harmonics on magnetic properties is discussed. Also it is demonstrated that the popular use of simple parameters characterizing texture like mean deviation angle or maximum deviation angle may give incorrect results. The problem of calculating the texture from the magnetic data is discussed.
1. Introduction The effective magnetic anisotropy in polycrystalline fine particle magnets arises not only from the magnetocrystalline anisotropy of the particles or from their shape but is strongly influenced by texture. The meaning of the word texture as used in this paper implies two pieces of information. Firstly the distribution of crystallographic directions and secondly the distribution of the particle orientation. Other forms of anisotropy like stress anisotropy surface or exchange anisotropy will be neglected. If the easy magnetic direction of a particle is determined by the magnetocrystalline anisotropy, information about the texture can be obtained f r o m X-ray or n e u t r o n diffraction experiments. If, however, the magnetic anisotropy of the particle is decided by its shape microscopical observations" provide the experimental data about the texture. These two distribution functions, different in their definition, m a y often be identical, for example an alnico magnet in a finely dispersed state consists of elongated ferromagnetic precipitations in a non-magnetic matrix. The crystallographic texture of this matrix determines the distribution of the easy magnetic axis.
There are examples of important materials where the texture is a decisive factor in obtaining the required magnetic properties; as such we can mention the alnico permanent magnets, S m C o 5 or BaFe12019. Also the orientation of magnetic particles is a continuing problem in the development of superior recording films. The mangetization processes in these magnets can be described assuming that each individual particle or grain constitutes a quasi-single domain. The easy axis of individual particles can be determined b y the magnets crystalline anisotropy or by the particle shape. Usually one type of anisotr o p y dominates. Magnetocrystalline anisotropy is for example, the p r e d o m i n a n t factor in C o - C r recording films [1] or in S m C o 5 magnets while the particle anisotropy shape dominates in most of the alnico alloys. In spite of its importance the influence of the texture on magnetic properties of single d o m a i n magnets has not been studied extensively. McCurrie [2] suggested that the alignment of the easy axis can be derived from remanence measurements. Higuchi and M i y a m o t o [3] c o m p a r e d the mean deviation angle in misorientation of crystals in alnico 8 with the calculated magnetic properties. M o o n [4] c o m p a r e d the measured mean
0 3 0 4 - 8 8 5 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
318
B. Szpunar et al. / Textured fine-particle magnets
deviation angle in alnico 5 specimens with the measured remanence and the coercive force. All these authors used the common assumption that the crystal orientation distribution is constant in the angular interval up to the mean deviation angle. Such an assumption does not agree with texture experiments and as we demonstrate later may lead to an error in the calculation of the initial susceptibility, coercive force and remanence. There is thus a necessity for more precise quantitative information about the texture. Such a quantitative description of the texture [5] is extensively used in order to explain the anisotropy of plastic and elastic properties of polycrystalline metals. Also the magnetic properties like magnetocrystalline energy [6] or magnetostriction [7] in textured materials can be calculated using the series expansion coefficients of the crystal orientation distribution function (ODF). A general description of texture is rather complicated, however, in many practical problems the symmetry of the orientation distribution and the magnetic symmetry of the particle simplifies the description. For the experimental problem discussed in this paper the texture may be described by the series expansion of Legendre's polynomials since in the majority of technical magnets, the orientation of the magnetic particles is symmetrical about the direction of the external magnetic field. The calculation will be carried out by assuming that the particle is a single domain at zero field and it will stay saturated at any field. The same mechanism of rotation of the spin in unison has been used by Higuchi and Miyamoto [3] giving agreement with the measurement of the coercivity. However, other mechanisms, particularly curling, may contribute to the results. Also the assumption will be made that the magnetic particles are dispersed, thus the interaction between them can be neglected.
2. Model
We dealt with an assembly of non-interacting, single domain particles. We assume that the particles' shape is ellipsoidal with the polar axis " O "
and equitorial axis " T " . Also they may have uniaxial magnetocrystalline anisotropy with an easy axis parallel to the " O " direction. For particles having a spherical shape the discussion will be limited to the effect of texture only. The particle magnetic anisotropy is considered to be the same which means here a constant composition or shape for all particles. It is usually the case that the composition is constant. Systems with the same shape are more difficult to find and one has to be satisfied with information about the mean value of the particle shape anisotropy. The assumption of the same magnetic anisotropy of the particle does not mean that the approach cannot be used to calculate the magnetic properties in systems having a variety of particle shapes or even different composition. Since the particles are non-interacting the rule of superposition applies and the magnetic properties can be calculated accordingly. If the particle shape is eUipsoidal and the easy axis does not coincide with the " O " axis, separate information on the distribution of the easy axis is necessary. This case seldom occurs and is not discussed in this paper. Usually one of these two anisotropies (texture or distribution of orientation of ellipsoidal particles) is dominant. The effect of it on the magnetic properties may be calculated by the proposed model. We introduce now two coordinate systems, one attached to the specimen (with three orthogonal axes, X, Y, Z ) and another, that of the particle (axes coincide with " O " and " T " directions). The direction of the magnetic field is defined by the polar angles q~, o in the frame of the specimen and by the angles a, fl in the reference frame of the particle. The orientation distribution function of the magnetic field vector, g+o(a, fl), is defined as the probability that H(qJ, o) coincides with the direction t~, fl in every particle. We take the magnetization to be in the direction M(0, @) (see fig. 1) and the external magnetic field to be in the direction H ( a , fl). The total energy is expressed in the specimen reference frame as a sum of the energies of the magnetocrystalline anisotropy E A, demagnetization energy E D and the energy Eia of the magnetic interaction between the magnetization and the
B. Szpunar et al. / Textured fine-particle magnets
z[
319
energy becomes:
I"I (~ o') H'(ct ~)
fly = ½(K sin2a + K sin2(90 - a ) ) =
K/2.
Fig. 1. Symbols.
This gives the wrong conclusion because if the average anisotropy is not dependent on a the m e a n magnetization can be directed in any direction. We should proceed with the minimization first which gives a m i n i m u m at a = 0 ° for the first particle and at 90 ° for the second particle. The calculated direction of the average magnetization is thus at a = 45 °.
field.
3. Initial susceptibility
~' (0, ¢)
t,,X
E = EA + E D + E H.
(1)
The magnetocrystalline anisotropy energy is given by EA =
K sin20
=
K - K cos20.
(2)
The demagnetization energy can be expressed as: E D = ½M2(N0 -
Nr)cosZO,
(3)
to simplify t h e presentation we introduce
F o r low magnetic field and uniaxial anisotropy minimization of the energy over 0 and q~ can be done analytically. Following Stoner and Wohlfarth, ref. [8] concerning an assembly of particles having shape anisotropy an analytical expression for the initial susceptibility can be derived. Minimization of the energy over ~ can be done for any value of the external magnetic field. The derivation of the energy with respect to ~ gives:
OE/Oq~=
7= ½M2(NT- No) + K
=
and then EA + E D =
- T cos20.
(4)
We should notice that in the expression of the energy the constant terms can always be omitted. The energy of an interaction with the external magnetic field H is expressed by: E H = -MH(sin
et cos/3 sin 0 cos q~
+ sin a sin/3 sin 0 sin ~ + cos 0 cos et).
O E H / O dp
-HMsinasinOsin(fl-qJ)=O,
(6)
which leads to the condition/3 -- ,/,. This result is crucial in obtaining a non-zero projection of the magnetization in the direction of the external magnetic field for a r a n d o m distribution function. Utilizing this condition one obtains the formula for the total energy. E = - T cos20 -
MH c o s ( a - 0).
(7)
Minimizing the energy we get (5)
W h e n attempting to find the formula for the average energy of textured particles we have to stress however, that the procedures of averaging and minimization do not commute. We m a y get wrong result if the E n is averaged over the orientation distribution function first. Consider for example two spherical particles, at a very low magnetic field. Both have uniaxial anisotropy, their easy directions being perpendicular to each other. The formula for the average
OE/OO= - 2 T cos 0 sin 0 - MH s i n ( a - 0) = 0.
(8) F o r the low values of the field strength, which means that IMH/2TI << 1, the magnetization vector is nearly parallel to the Z-axis of the specimen reference frame. Also, as sin 0 << cos 0, it follows therefore that sin 0 = M H ( s i n
a)/2T.
(9)
The projection of the magnetization on the exter-
B. Szpunar et al. / Textured fine-particle magnets
320
hal magnetic field is
M H = M ~ cos(a - O)g¢o(ct , fl)sin a d a dfl. (10) By use of eq. (9) we can proceed now with the calculation of the average magnetization. One finally obtains
Mn=
M 2 H f~ f 2 ~ . 3 2T Jo Jo sinagg, o(a, fl) d a d f l +
~o ~o
1
8T 2
x sin 2a g~o(a, fl) d a dfl.
(lOa)
Because, the orientation distribution function is of an even power with respect to cos a the second part of eq. (lOa) is equal to zero and the following formula for the initial susceptibility is obtained OM~
x
aH._o
M 2
/'~ /'2~r
. 3
results as those of Stoner and Wohlfarth [1]. The derived expression of the susceptibility is also valid for particles having cubic magnetocrystalline anisotropy since for low values of 0 the cubic anisotropy and uniaxial anisotropy are expressed by a similar formula. The use of formula (11) to calculate the susceptibility requires the quantitative representation of the texture. All magnetic parameters are dependent only on the angle a which means that the texture function g~o(a, fl) may be simplified. The general form of the texture function is
"
oo
M(I)
g~o( a, fl) = E
E t'~k~'(a, ill,
(12)
I~0 m=l
where surface spherical harmonics k T ( a , fl) are expressed using associated Legendre functions P~(cos a), namely
TTJoJo slnag¢o(a, fl)dadfl. (11)
For the random distribution (a, fl) = 1 / 4 ~ one obtains
function
g+o
X = ~M2/T. If additionally the magnetocrystalline anisotropy can be neglected ( K = O) one obtains the same
k'/'(a, fl) - ~~
e " ~ -Pt" " (cos ,~).
(13)
Since the texture used for magnetic calculations is only an a function, the measured texture g~o(a, fl) can be integrated over fl or an experiment can be performed which gives automatically the required integration. The texture will thus be described by the ex-
Table t C o n s t a n t s u s e d in the c a l c u l a t i o n o f the i n f l u e n c e o f the t e x t u r e o n the m a g n e t i c p r o p e r t i e s Legendre polynomials
Texturesusceptibility
Textureremanence
Texturecoercivity
Textureanisotropy
Pt(a)
coefficient
coefficient
coefficient
field
Xl
vl
8ct
~Ai
P0 ( a )
0.943
0.707
0.530
1.672
P2 ( a )
- 0.422
0.395
0.215
- 0.145
P4 ( a )
0
- 0.088
- 0.082
0.223
Pr(a)
0
0.040
0.077
- 0.061
Ps ( a )
0
- 0.023
- 0.011
0.032
Plo(a)
0
0.015
0.020
- 0.035
P12(a)
0
-- 0 . 0 1 0
- 0.005
0.051
Pt4 ( a )
0
0.008
0.013
-- 0.023
Pa 6 ( a )
0
- 0.006
- 0.002
0.033
Pls(a)
0
0.005
0.007
- 0.017
P20 ( a )
0
- 0.004
-- 0.002
0.024
B. Szpunar et aL / Textured fine-particle magnets
pression L
g~o(a) = E t , P t ( a ) -
(14)
/=0
From (11) and (14) we get M
X=-~
2
L
F_, I=0
ttf
,n
sin 3a fit(a) d a .
321
is in the direction closest to the magnetic field gives us maximum and minimum values of the remanence. M R for particles having uniaxial anisotropy is theoretically between 0 and 1. For cubic spherical particles having easy direction [100] or [111] the changes of the remanence due to the texture are limited to the interval
(15)
1/7c3
*'0
Introducing a texture-susceptibility coefficient )kt = ~ s i n 2 a .P/(a) d a ,
(16)
<
MR/M s
<
1.
(20)
In materials which have [110] easy direction this interval is smaller, namely
V/~ < M R / M s < 1.
(21)
we obtain M2 ±
X = ~-~ )'~ ttX,.
(17)
I=0
Assuming that the texture function is given by formula (14) we can modify the expression (19) and obtain: L
The values of Xt are given in table 1. Substituting the calculated value for P2(a) one obtains
MR = E tt`/t,
M2 X = - ~ - (0.943to = 0-422t2 ).
where 39 is given by
(18)
where t 0, t 2 are the texture expansion coefficients (t o = 1/v~-). All other integrals over higher order polynomials are zero.
Ms
(22)
t=0 ,ff
"// = ½fo P t ( a ) s i n 2 a d a .
(23)
Calculated values of "/t are given in table 1.
4. Remanence 5. Coercivity Frei et al. [9] suggested the formula for the relationship between the remanence and the distribution of the easy axes of uniaxial ferromagnetics for an assembly of single domain particles. Bunge [5] also used the texture formalism and derived the expression for the remanence in textured hexagonal metals. A similar approach will be used here in order to obtain the coefficients which m a y be used directly to calculate the remanence in textured single domain magnets. The mean relative component of magnetization in the direction of the original field can be calculated from the following formula:
MR:f2. Ms - - ' o "o cos a g,~o(a, fl)sin a d a
We report here Stoner's and Wohlfarth's [8] calculation including additionally the anisotropy described by the orientation distribution function. For the coercive field the following conditions are valid: M c o s ( a - 0) = 0,
0 e / 0 o = o, ~.2E/002 > O.
(25)
These conditions are fulfilled only when a > ~r/4. In this case the coercivity is given by:
H c = ( 2 T / M ) s i n a cos a, dfl,
(26)
(19)
where M R is the remanence and M s is the saturation magnetization. The condition that the domain
and for textured materials this becomes,
•/c=
2T : ' e2~r
-joJ0
• 2
g¢o(a, f l ) s m a c o s a d a d f l .
(27)
322
B. Szpunar et al. / Textured fine-particle magnets
One should stress, here however, that the averaging of the coercivity over the texture function is only an approximation [10], because the assumption of homogeneous magnetization is not fully justified. For a < ~ / 4 the coercivity is equal to the multivalue critical field H M. The conditions for the anisotropy field are in this case O E / O 0 = 0 and OZE/O02 = 0.
(28)
Following Stoner's and Wohlfarth's calculations we have: 2T HM =--~-V/1 -- tz + t4 / ( t2 + 1 ) ,
(29)
3 where t = ~ t g a . For textured materials one obtains 2T fo~fo2 ~rv/1 - t2
+
t4 /(
+1)
× g ¢ o ( a , fl)sin a d a dfl.
(30)
According to eqs. (14), (30) and (4) the coercivity for a symmetrical orientation distribution will be given by
/4c
2T
(31)
tirol, I=O
where the texture coercivity coefficients are defined by |
2] U2 t( )sin2 [~,~/4
+ f0~/4~/1
-
cos do
t 2 + t 4 / ( t 2 + 1)
× P/( )sin
(32)
The anisotropy field H A = 2 H M is given by
4s
HA = 2HM = M
tt~a"
(33)
/=o where 8Ai equals 8A, = f0~V/1 -- t 2 + t 4 / ( t 2 + 1) PI (or)sin et dc~. (34) The texture-coercivity constant 8c~ and the anisot-
ropy constant 8Al are given in table 1 for the different values of I.
6. Discussion
The magnetization in a powder magnet subject to a uniform external field varies from particle to particle, also the internal field on each particle is different due to its own magnetization and the magnetization of neighbouring particles. Since the assumption is made that particles do not interact, the internal field is considered uniform through the magnet and the calculation of the mean magnetization is justified. This simplified assumption is used for calculation of the mean value of the initial susceptibility and the remanence. The calculation of the mean coercivity has been made by averaging the coercivity over the texture function. Such method is justified only if the magnetization throughout the magnet is homogeneous. It has been used previously by other authors, i.e. Lawton and Stewart [11]. The average value of the coercive field obtained using this method for a random texture is 0.375. The result does not agree with the value H c = 0.479 obtained by Stoner and Wohlfarth for a random assembly of particles and a mean value of the magnetization. These two approaches to the calculation of the coercive force are compared in fig. 2 with the experimental results obtained by Paine, Mendelsohn and Luborsky [12]. Our approach is closer to the experimental data obtained for single domain iron particles. Both results represent the two extreme approaches to the problem of averaging the magnetic properties and neither the assumption of zero field from the neighbouring particles nor that of uniform magnetization is fully justified. The calculation of the susceptibility, the coercivity and the remanence in single domain textured magnets requires the knowledge of series expansion coefficients characterizing the texture. The number of coefficients which are necessary is influenced by the orientation distribution function characterizing the examined specimen but is more
323
B. Szpunar et al. / Textured fine-particle magnets I
I
I
I
I
I
t~000
II~
/ ./t
_
,/
f ~LCALCULATEDBY sTONERAND
,~
3000
/ /
uJ"
/
/
,92000_
--
WOHLFARTH~ f
/
/// /
-
PRESENT METHOD
oBsj._~ +
I
I 2
1
I
I 3
L
I t~
1
MEDIAN LENGTH TO DIAMETER RATIO Fig. 2. Relation between medium elongation and coercive force
by Paine, Mendelsohn and Luborsky [12] compared with our calculation.
strongly d e p e n d e n t o n the magnetic property which is calculated. A n a l y s i n g the result given i n table 1 we see that for the calculation of the initial susceptibility only two texture coefficients are required. The texturesusceptibility c o n s t a n t s for l > 4 are all zero. T h e "tz c o n s t a n t s characterizing the value of the reman e n c e decrease with a n increase in l a n d o n l y the
first few texture coefficients are needed. The function describing the coercive field H c a n d the anisotropy field H A is more complex, a n d therefore a higher n u m b e r of associated Legendre p o l y n o m i a l s is required for the calculation of the a n i s o t r o p y field in textured materials. The relationship between the texture a n d the a n i s o t r o p y of m a g n e t i c properties is illustrated b y c o l u m n a r alnico magnets. Texture data are taken from the p a p e r b y M o o n [4]. Histograms illustrating the dispersion of [001] direction a b o u t the m a g n e t axis for the 8 different specimens are analysed a n d the series coefficients characterizing the texture i n these specimens are calculated. Three specimens have their o r i e n t a t i o n distribution f u n c t i o n s in the interval 0 ° - 1 5 ° . F o r the r e m a i n i n g specimens the d i s t r i b u t i o n is slightly broader, however, o n l y negligible intensity is observed at angles higher than 20 ° . F o r the eight analysed o r i e n t a t i o n d i s t r i b u t i o n s the changes in the t 2 texture e x p a n s i o n coefficient is a b o u t 5% of its m e a n value. The higher order coefficients c h a n g e d more strongly i.e. t 4 changes b y a b o u t 20% 16 a b o u t 40% a n d t 8 more t h a n 80%. T h e calculations showed (see e.g. table 2) that the texture most significantly influences the initial susceptibility; the variation in a m p l i t u d e of the coefficients characterizing the texture of the vari-
Table 2 Magnetic parameters characterizing specimens of alnico 5. Remanence and coercivity as measured by Moon [4]. The magnetic parameters as calculated in this paper using the series expansion method for l = 8, 1= 20 and directly for the measured histogram characterizing texture (exp) Specimen
1A
1B
2A
5
Series
Calculated
expansion
susceptibility
remanence
coercivity
anis. f i e l d
Measured
l
x/(M2/2Ke)
dr/Js
H¢/(2Ke/M)
HA/(2K=/M )
8 20 exp 8 20 exp 8 20 exp 8 20 exp
0.040 0.040 0.040 0.028 0.028 0.028 0.018 0.018 0.018 0.016 0.016 0.016
0.969 0.979 0.979 0.972 0.985 0.986 0.973 0.990 0.991 0.973 0.996 0.992
0.665 0.693 0.696 0.676 0.708 0.707 0.686 0.734 0.733 0.688 0.742 0.747
1.380 1.388 1.385 1.414 1.415 1.414 1.452 1.470 1.467 1.460 1.490 1.494
remanence (T)
coercivefield Hc (kA/m)
1.34
55.7
1.34
57.3
1.35
61.7
1.34
59.7
324
B. S z p u n a r et al. / Textured f i n e - p a r t i c l e magnets
ous specimens may change the initial susceptibility by a factor of three. Unfortunately we do not have the experimental data for the susceptibility of the analysed specimens. However, similar changes in the initial susceptibility for alnico alloys have been measured by Higuchi and Miyamoto who demonstrated that the susceptibility changes from 0.024 to 0.092 G / O e (table 3) correspond to a change of the mean deviation angle from 22 ° to 45 ° . The same authors have estimated the deviation angle from the measured susceptibility for two specimens which have the same value of remanence. From the estimated deviation angle they calculated the remanence and as we see in table 3 (specimens 9 and 10) they obtained results which completely disagree with the experimental values of the remanence. This example illustrates the fact that the use of the average deviation angle for characterizing the texture is unsatisfactory. Since susceptibility-texture constants "/i are expressed in terms of sin2et any change in the texture at low angles may produce significant changes in the initial susceptibility. Such changes frequently occur (see fig. 3) but cannot be adequately described using the mean deviation angle. Description of the shape of the texture function using series expansion coefficients offers the possibility of precise calculations. It has already been demonstrated that the initial susceptibility can be calculated knowing only one texture coefficient t 2 since the "//for l > 2 are zero (table 1). For the investigated specimens we observe significant changes in the amplitude of higher order expansion coefficients. Such changes may only affect other magnetic properties such as the remanence, the coercive field and the anisotropy field. The values of the remanence calculated for
20 18 16 1/*
-t=50 --t:36 ....... t=20
12 10 ~:
8
~
----t:8
/
.....
=
2
6 /, 2 0
50
100
150
200
250
Fig. 3. Histogram of texture for 1A specimen. Functions illustrating the series expansion approximation for / = 2, 8, 20, 36 and 50 are added.
analysed specimens do not change significantly (less than 1%). As we see from the definition of the "// coefficients (eq. 23) changes of cos et in the interval (0°-15 °) are small and thus coercivity is not sensitive to changes of orientation density in this interval. It is interesting to note that McCurrie [2] suggested that the extent of the easy axis alignment in uniaxial permanent magnets can be calculated from remanence measurements. We believe that in most technical magnetic materials which exhibit small differences in texture it will be difficult and inaccurate, even if measurements are made in the direction perpendicular to the axis of alignment. As an example (see table 2) the remanence measured in the specimens 1A, 1B and 5 is the same, our calculation also agrees with these experimental results, howex/er the initial susceptibility
Table 3 Results obtained by" Higuchi and Miyamoto [3] Specimen no.
9 10
Observed magnetic properties
Calculated magnetic properties
Deviation angle
Br
H~
X
Br
Hc
Om
(kG)
(kOe)
(G/Oe)
(kG)
(kOe)
(deg)
11.6 11.6
1.52 1.46
0.024 0.066
11.6 10.2
1.68 1.47
2215 37 26
325
B. Szpunar et al. / Textured fine-particle magnets
varies from 0.16 to 0.40. The formula suggested in ref. [2] was derived assuming that there is a constant density of the distribution of the easy axis up to the maximum misorientation angle. As we demonstrated, such an assumption is oversimplified and we suggest the solution which makes use of the experimentally measured orientation distribution. Our results demonstrate that initial susceptibility is most sensitive to texture changes in technical alnico magnets. However its use to determine the texture quantitatively is limited, since it is only sensitive to changes in the t 2 texture coefficient. The same applies to the coercive force, because several texture series expansion coefficients have to be known in order to describe the crystal orientation distribution. The coercive force in the investigated specimens is dependent on the shape of the crystal orientation distribution. The changes of texture in the analysed specimens are, according to the theory, responsible for a 5% variation in the coercive force. Fig. 4 illustrates the relation between the experimental value of the coercive force and the amplitud.e of the texture series expansion coefficients. Contrary to its effect on the initial susceptibility an increase in the amplitude of the second order texture coefficient causes an increase of the coercive force. The changes in the amplitude of the higher order series expansion coefficients contribute significantly to the theoretical value obtained for the coercive force. The contribution of observed changes in the value of t 2 is roughly the same as the contribution of changes in the t 4 coefficient. Changes in the t 6 texture coefficient contribute the most to the observed differences in the coercive field of the analysed specimens. The calculated values for the coercivity (see table 2) are however dependent on higher order texture coefficients. A series expansion up to l = 14 is, for the analysed specimens, sufficient. Termination of the series expansion at l = 14 may however not be sufficient for describing texture, especially in the case of a strong texture where the orientations of crystals are concentrated in a small angular interval. As an illustration (fig. 3) we compare the shape of the function termination at the different values of 1, l = 8, 20, 36 and 50. The series expansion
/
60
t (el t (+) f6(o} t8 (~x)
"6
<:
~ / fi
f .,*
/
59
/
58
/ /
i
I
I
I
t
I
I
12
1~
16
1-8
20,
22
AMPLITUDE OF TEXTURE SERIES EXFANSIONCOEFFIEIENT$
Fig. 4. E x p e r i m e n t a l c o e r c i v e f o r c e as a f u n c t i o n o f t h e a m p l i t u d e o f t h e t e x t u r e series e x p a n s i o n c o e f f i c i e n t s f o r a l n i c o 5.
l = 50 describes the main features of the experimental distribution, however its maximum is lower than that of the experimental histogram.
7. Conclusions Describing the initial susceptibility, the remanence, the coercive field and the anisotropy field using the same orthogonal functions which are used in the description of the texture makes it possible to discuss and to analyse the influence of the texture of the magnetic properties. The formulae for the calculations of these properties in single domain fine particle magnets have been suggested. The coefficients ~1, 71, 8ct, 8 A I have been introduced which characterize the influence of the texture on the various magnetic properties. Accordingly the initial susceptibility is dependent only on one texture coefficient t 2. To calculate other magnetic properties we need however more precise information about the texture and the texture coefficients up to ! = 14 may be needed in the case of alnico type alloys. The remanence in these alloys is not strongly dependent on the observed texture variation. Initial susceptibility however is very sensitive to texture changes. The observed changes in the coercive force with the texture are small, 5%. The quantitative description of texture which
326 has b e e n i n t r o d u c e d has a n descriptions. For example d e v i a t i o n a n g l e m a y l e a d to the initial s u s c e p t i b i l i t y a n d
B. Szpunar et al. / Textured fine-particle magnets
advantage over other the use of the m e a n a n i n c o r r e c t v a l u e for remanence.
Acknowledgements T h e a u t h o r s w i s h to t h a n k D r . B.K. T a n n e r for v a l u a b l e d i s c u s s i o n s a n d are i n d e b t e d to Dr. W . D . C o r n e r for c o m m e n t s o n t h e m a n u s c r i p t . W e g r a t e f u l l y a c k n o w l e d g e the c a r e w i t h w h i c h Mrs. V. T o d d a n d M r s . M . B r a d l e y t y p e d the manuscript.
References [1] s. lwasaki and K. Ouchi, IEEE Trans. Magn. MAG-14 (1978) 849.
[2] R.A. McCurrie, J. Appl. Phys. 52 (9181) 7344. [3] A. Higuchi and T. Miyamoto, IEEE Trans. Magn. MAG-6 (1970) 218. [4] J.R. Moon, J. Phys. D7 (1974) 1233. [5] H.J. Bunge, Texture Analysis in Materials Science (Butterworths, London, 1982). [6] J. Szpunar and M. Ojanen, Met. Trans. 6A (1975) 561. [7] J. Szpunar, J. Appl. Phys. 55 (1984) 2133. [8] E.C. Stoner and E.P. Wohlfarth, Phil. Trans. Roy. Soc. A240 (1948) 599. [9] E.M. Frel, S. Shtrikman and D. Treves, J. Appl. Phys. 36 (1953) 443. [10] E.P. Wohlfarth, private communication. [11] H. Lawton and K.M. Stewart, Proc. Phys. Soc. 63A (1950) 848. [12] T.O. Paine, L.I. Mendelsohn and F.E. Luborsky, Phys. Rev. (1955) 1055.