- Email: [email protected]
Dependence of permeability spectra on microstructure
D E P E N D E N C E OF P E R M E A B I L I T Y S P E C T R A ON M I C R O S T R U C T U R E OM P R A K A S H , R. A I Y E R and C. M. S R I V A S T A V A Department of Physics, Indian Institute of Technology, Bombay-400076, India
It is shown that in polycrystalline pure and AI- and Gd- substituted garnets, a resonance type permeability spectrum is obtained for small grains where closure domains are not energetically possible, while a relaxed type spectrum is obtained for large grains having closure domains.
1. Introduction
The permeability spectra of ferrites and garnets have been studied extensively in the past [1, 2]. However, the dependence of the shape of the permeability spectra on microstructure has not been studied in detail. The attempt by Globus [3], to explain the resonance and relaxation types of spectra arising due to uniform and non-uniform grain diameters respectively, is not in agreement with our observations. In the present work we have studied the permeability spectra of pure and A1- and Gd-substituted polycrystalline garnets with 47rM s varying from 80-1800 G and grain size varying from 1-20/~m. Samples with densities greater than 98.5% X-ray density were prepared using hot pressing and conventional sintering techniques. Our observations indicate that the grain size D, is the dominant factor in determining the shape of the permeability spectrum. For the same composition resonance spectra have been observed for small D < Dcrit , and relaxed spectra for large D > Dcrir For D D¢n t mixed type spectrum has been observed. A representative spectrum for each of these three types for pure Y I G is given in figs. la, lb and lc for grain diameters 1.9, 6.0 and 8.5/xm respectively. 2. Analysis and discussion
The observed grain size dependence of the shape of the permeability spectrum has been analysed on the basis of the domain structure within a grain. It is shown that the resonance spectrum appears when closure domains are absent for D < Dcrit while the relaxed spectrum occurs when closure domains are possible for D > Den t. We calculate the number of domain walls, N, in a spherical grain of diameter D, considering 180 ° domain walls only. The total energy, E, consists of the wall energy and the magnetostatic energy.
Minimizing E with respect to N, it has been shown [4] that the number of domain walls in a grain of diameter D is given by
N=
27r MiD 3
1
-1
(1)
ow
where, M s is the magnetization and o w is the wall energy per unit area. For N = 1 the grain diameter, D, from eq. (1) is (15Ow/2~rM~). Since there is only one domain wall within the grain of this diameter closure domains cannot be formed. To obtain the condition for the formation of closure domains in Y I G we need to consider the relative contributions of the magnetostatic, E,1, and the magnetoelastic, Ex, energies [51. For small N, the magnetostatic energy can be shown [7] to vary as E m -- 0 . 5 5 o 3 m 2 [ 2 / / ( N
+
1)].
(2)
Assuming magnetostriction (h) to be isotropic the magnetoelastic energy density can be written in the form [6]. e =
sin20
(3t
where T is the tensile stress and 0 is the angle between M s and T. We assume that the closure domain will form when the magnetostatic energy exceeds the magnetoelastic energy for the equilibrium number N o of the domain walls given in eq. (1). The volume occupied by the closure domains in a spherical grain is difficult to calculate. We estimate it on the basis of prismatic domains in a cube of volume 1 cm 3 which is about 5% for iron. Taking it to be approximately the same for Y1G and taking 0 = ~r/2, we obtain the following condition for the formation of closure domains,
0.55M212/(N+
1)] = 0.04hT.
(4)
In the case of substituted garnets it is assumed to vary linearly with M s. We then have M s = 0.035
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1521-1522 @North Holland
()~oT//Mo)(No +
1),
(5) 1521
1522
O, Prakash et a l . / Dependence of permeability spectrum on microstructure M• ~F,~/~ ¸
~
C
q
I
q
Lt.~
-6L
]
r' "tl
tL
! i i 0 /
:i
,
1
,
,
N
i ~
~
T
Fig. 2. Nature of permeability spectrum as a function of N, number of domain walls within a grain and M s. The line AB indicate, the transition boundary for change from the resonance to the relaxed spectrum and is given by eq. (5).
100
t
a~
G~
where A0 and M 0 are respectively the magnetostriction constant and magnetization of pure YIG. From eq. (1) and eq. (5) we obtain
~n I£0
\ Dcri t F0
f ~Hz F 1bL
1
I I
"\?
'aOb
"
\
i
o
¥1C, ~",,
2
I;0,
Ms
/
~00
3 Ow 2~" M~
1+
A0T
•
(61
In fig. (2) is plotted the nature of the observed spectrum as a function of N and M~. The line AB represents the observed transition boundary for the change from resonance to relaxed spectrum as predicted by eq. (5). From the slope of AB taking A0 for YIG as 1.9 × 10 - 6 , T is observed to be 2.2 x 10 m dynes cm-2. The theory satisfactorily accounts for the observed dependence of the nature of the permeability spectrum on M~ and grain size.
¥
16,~
_
a b ]LLm I . U ~mu,,
,
~d
110 ` o
References
8o
6O
--~0
//Z/!,.o
Lo /
20
0 L lo ~
, 100 ~
~
, I.
Io ~
.i_J o 100.
Fig. 1. Permeability spectra of YIG for samples with grain diameter (a) 1.9 ~m, Co) 6 ~.m, and (c) 8.5 ~m. The spectrum in (a) is resonance type, in (c) is relaxed type, and in (b) it has a mixed character.
[1] O. T. Rado, R. W. Wright and W. H. Emerson, Phys. Rev. 80 (1950) 278. [2] J. Verveel, Proc. IEE, B109, Suppl. 21 (1962) 95. [3] A. Globus, Conf. on Soft Mag. Materials 2, Gardiff, European Physical Society (1975). [4] Om Prakash, R. Aiyer and C. M. Srivastava, Mater. Sci. Bull. 1 (1979)49. [5] J. F. Dillon Jr., in: Magnetism Vol. III, eds. G. T. Rado and H. Suhl (Academic, N e w York, 1963) p. 415. [6] C. Kittel and J. K. Galt, in: Solid State Phys., Vol. 1II, eds. F. Seitz and D. Turnbull (Academic, New York, 1956) p. 437. [7] C. M. Srivastava, Om Prakash and R. Aiyer (to be published).
It is shown that in polycrystalline pure and AI- and Gd- substituted garnets, a resonance type permeability spectrum is obtained for small grains where closure domains are not energetically possible, while a relaxed type spectrum is obtained for large grains having closure domains.
1. Introduction
The permeability spectra of ferrites and garnets have been studied extensively in the past [1, 2]. However, the dependence of the shape of the permeability spectra on microstructure has not been studied in detail. The attempt by Globus [3], to explain the resonance and relaxation types of spectra arising due to uniform and non-uniform grain diameters respectively, is not in agreement with our observations. In the present work we have studied the permeability spectra of pure and A1- and Gd-substituted polycrystalline garnets with 47rM s varying from 80-1800 G and grain size varying from 1-20/~m. Samples with densities greater than 98.5% X-ray density were prepared using hot pressing and conventional sintering techniques. Our observations indicate that the grain size D, is the dominant factor in determining the shape of the permeability spectrum. For the same composition resonance spectra have been observed for small D < Dcrit , and relaxed spectra for large D > Dcrir For D D¢n t mixed type spectrum has been observed. A representative spectrum for each of these three types for pure Y I G is given in figs. la, lb and lc for grain diameters 1.9, 6.0 and 8.5/xm respectively. 2. Analysis and discussion
The observed grain size dependence of the shape of the permeability spectrum has been analysed on the basis of the domain structure within a grain. It is shown that the resonance spectrum appears when closure domains are absent for D < Dcrit while the relaxed spectrum occurs when closure domains are possible for D > Den t. We calculate the number of domain walls, N, in a spherical grain of diameter D, considering 180 ° domain walls only. The total energy, E, consists of the wall energy and the magnetostatic energy.
Minimizing E with respect to N, it has been shown [4] that the number of domain walls in a grain of diameter D is given by
N=
27r MiD 3
1
-1
(1)
ow
where, M s is the magnetization and o w is the wall energy per unit area. For N = 1 the grain diameter, D, from eq. (1) is (15Ow/2~rM~). Since there is only one domain wall within the grain of this diameter closure domains cannot be formed. To obtain the condition for the formation of closure domains in Y I G we need to consider the relative contributions of the magnetostatic, E,1, and the magnetoelastic, Ex, energies [51. For small N, the magnetostatic energy can be shown [7] to vary as E m -- 0 . 5 5 o 3 m 2 [ 2 / / ( N
+
1)].
(2)
Assuming magnetostriction (h) to be isotropic the magnetoelastic energy density can be written in the form [6]. e =
sin20
(3t
where T is the tensile stress and 0 is the angle between M s and T. We assume that the closure domain will form when the magnetostatic energy exceeds the magnetoelastic energy for the equilibrium number N o of the domain walls given in eq. (1). The volume occupied by the closure domains in a spherical grain is difficult to calculate. We estimate it on the basis of prismatic domains in a cube of volume 1 cm 3 which is about 5% for iron. Taking it to be approximately the same for Y1G and taking 0 = ~r/2, we obtain the following condition for the formation of closure domains,
0.55M212/(N+
1)] = 0.04hT.
(4)
In the case of substituted garnets it is assumed to vary linearly with M s. We then have M s = 0.035
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1521-1522 @North Holland
()~oT//Mo)(No +
1),
(5) 1521
1522
O, Prakash et a l . / Dependence of permeability spectrum on microstructure M• ~F,~/~ ¸
~
C
q
I
q
Lt.~
-6L
]
r' "tl
tL
! i i 0 /
:i
,
1
,
,
N
i ~
~
T
Fig. 2. Nature of permeability spectrum as a function of N, number of domain walls within a grain and M s. The line AB indicate, the transition boundary for change from the resonance to the relaxed spectrum and is given by eq. (5).
100
t
a~
G~
where A0 and M 0 are respectively the magnetostriction constant and magnetization of pure YIG. From eq. (1) and eq. (5) we obtain
~n I£0
\ Dcri t F0
f ~Hz F 1bL
1
I I
"\?
'aOb
"
\
i
o
¥1C, ~",,
2
I;0,
Ms
/
~00
3 Ow 2~" M~
1+
A0T
•
(61
In fig. (2) is plotted the nature of the observed spectrum as a function of N and M~. The line AB represents the observed transition boundary for the change from resonance to relaxed spectrum as predicted by eq. (5). From the slope of AB taking A0 for YIG as 1.9 × 10 - 6 , T is observed to be 2.2 x 10 m dynes cm-2. The theory satisfactorily accounts for the observed dependence of the nature of the permeability spectrum on M~ and grain size.
¥
16,~
_
a b ]LLm I . U ~mu,,
,
~d
110 ` o
References
8o
6O
--~0
//Z/!,.o
Lo /
20
0 L lo ~
, 100 ~
~
, I.
Io ~
.i_J o 100.
Fig. 1. Permeability spectra of YIG for samples with grain diameter (a) 1.9 ~m, Co) 6 ~.m, and (c) 8.5 ~m. The spectrum in (a) is resonance type, in (c) is relaxed type, and in (b) it has a mixed character.
[1] O. T. Rado, R. W. Wright and W. H. Emerson, Phys. Rev. 80 (1950) 278. [2] J. Verveel, Proc. IEE, B109, Suppl. 21 (1962) 95. [3] A. Globus, Conf. on Soft Mag. Materials 2, Gardiff, European Physical Society (1975). [4] Om Prakash, R. Aiyer and C. M. Srivastava, Mater. Sci. Bull. 1 (1979)49. [5] J. F. Dillon Jr., in: Magnetism Vol. III, eds. G. T. Rado and H. Suhl (Academic, N e w York, 1963) p. 415. [6] C. Kittel and J. K. Galt, in: Solid State Phys., Vol. 1II, eds. F. Seitz and D. Turnbull (Academic, New York, 1956) p. 437. [7] C. M. Srivastava, Om Prakash and R. Aiyer (to be published).