Materials Transactions, Vol. 44, No. 12 (2003) pp. 2594 to 2598
Special Issue on Structural and Functional Control of Materials through Solid-Solid Phase Transformations in High Magnetic Field
#2003 The Japan Institute of Metals
Characteristics of Paramagnetic and Diamagnetic Anisotropy
which Induce Magnetic Alignment of Micron-Sized Non-Ferromagnetic Particles
Chiaki Uyeda, Kenta Tanaka* , Ryouichi Takashima* and Makoto Sakakibara*
Institute of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan
The effect of temperature is discussed on the magnetic-alignment process of micron-sized particles dispersed in a fluid medium, based on
the experimental data compiled on various non-ferromagnetic materials having different concentrations of paramagnetic impurity ion. The fieldintensity required to achieve alignment decreased with temperature following the relation calculated from the Langevin theory, when the
diamagnetic particles were free of paramagnetic ions. The rotational Brownian motion was considered to randomize the direction of the microcrystals in the theory. The above-mentioned temperature dependence was expected to occur for most of the diamagnetic oxides, since the oxides
were expected to posses a finite amount of diamagnetic anisotropy according to a model proposed recently to explain the origin of anisotropy.
The decease of temperature caused additional reduction on the field-intensity to achieve alignment, when finite amount of paramagnetic ion was
contained in the particle. This was because the paramagnetic anisotropy increased which the reduction of temperature. The doping of
paramagnetic ion on non-ferromagnetic materials in the course of processing a material expected to reduce the field intensity to achieve
magnetic alignment at room temperature. The above findings, concerned with the reduction of field intensity to achieve magnetic alignment,
may increase the possibility of practical applications of the phenomena of magnetic alignment.
(Received June 20, 2003; Accepted November 14, 2003)
Keywords: diamagnetic anisotropy, magnetic alignment at low magnetic field, temperature dependence of magnetic alignment, magnetic
alignment of micron-sized particle, magneto-rotation, Curie temperature dependence of paramagnetic anisotropy, doping of
paramagnetic ion, ceramic material, kaolinite, graphite
1.
Introduction
It is known that micron-sized non-ferromagnetic particles
dispersed in a fluid medium generally possess an efficiency of
magnetic alignment due to its diamagnetic or paramagnetic
anisotropy .1,2) The magnetically stable axis of a particle
may align nearly parallel to the field direction, when the fieldinduced anisotropy energy exceed the energy of rotational
Brownian motion supplied from thermal motions of the fluid
molecules. The process of the alignment depend on three
parameters, namely the mol number of the particle N,
magnetic anisotropy of the material per mol and
temperature T. The effect of these parameters has not been
studied systematically as yet, in spite that the controlling of
the alignment process by means of these parameters is an
important basis to realize alignment at low field intensity.
The alignment at practical low field-intensity may produce
new types of applications based on the above-mentioned
alignment. Up to now, alignment has been studied mainly on
biological3) and organic materials4) in a strong magnetic field
above several Tesla.
Magnetic alignment of inorganic materials have been
performed on micro-crystals of some nonmetals, such as
kaolinite, talc,5) lepidolite, fluoro-phlogopite, muscovite,6)
graphite,7) Al2 O3 .8,9) and various ceramic materials.2) The
number of reports on inorganic materials, nevertheless, were
not large because their diamagnetic anisotropy ðÞDIA were
considered to be negligibly small compared to those of the
organic materials.10) The small ðÞDIA values of basic
inorganic oxides were detected by a new method developed
by one of the authors.11,12) The ðÞDIA values were obtained
for forsterite,12) corundum, muscovite, Mg(OH)2 , Al(OH)3 ,
AlOOH,13) orthoclase, apophylite,14) gypsum, KDP and
*Graduate
Student, Osaka University
ADP.15) The compiled data indicated that many of the
unmeasured diamagnetic-oxides may posses a finite amount
of anisotropy to cause magnetic alignment.16) Experimental
evaluations of the effects of the parameters which control the
magnetic alignment process, discussed above, can be
examined quantitatively by adopting the inorganic materials
in the alignment experiments.5–7) It was difficult, for
example, to alter the value of temperature over a wide range
in the experiment using a biological material.
The effect of temperature is discussed in the present paper
on the alignment of diamagnetic and paramagnetic particles,
based on the compiled data on inorganic materials.5,7,13,19,20)
It is seen that the reduction of temperature is effective in
lowering the field-intensity to achieve magnetic alignment
for most of the diamagnetic and paramagnetic particles. This
effect is caused by the temperature dependence of the
paramagnetic anisotropy ðÞPARA deriving from the impurity ions, and also by the decrease of the rotational
Brownian motion.7)
2.
Effects of the Parameters that Control the Magnetic
Alignment
The relationship between the degree of preferential alignments of the magnetically stable axes of the micron-sized
particles and the field-intensity B has been analyzed
consistently for various experiments,1,2) based on a method
first proposed by Langevin and Curie.17) The degree of
alignment was calculated from the Boltzmann distribution of
the micron-sized particles at a given temperature T. The
field-induced free energy U of a particle having a mol
number N was used in the calculation,
UðÞ ¼ ðNB2 =2Þf? þ cos2 g:
ð1Þ
The angle between the magnetically stable axis of the particle
Order Parameter <m>
Characteristics of Paramagnetic and Diamagnetic Anisotropy which Induce Magnetic Alignment of Micron-Sized Non-Ferromagnetic Particles 2595
alignment is almost completed, in order to carry out
quantitative comparison between different alignment processes. It is useful to define such a value in investigating the
quantitative amount of field intensity required for grain
alignment in the processes of various applications as well.
The field intensity where hmi reaches hmi ¼ 0:8 was defined
tentatively as BS for this purpose.5–7) The value of BS is
calculated from the Langevin theory as,
1
kaolinite
talc
graphite
BS ¼ ð15kB T=NÞ1=2 :
0
0.5
1
1.5
2
2.5
Magnetic Field B/ T
Fig. 1 Magnetic alignment process measured for various ceramic microcrystals dispersed in liquid ethanol.
and the field direction is denoted as in the above equation.
The magnetic anisotropy of the material composing the
material is described as ¼ k ? ; k and ? denote the
molar magnetic susceptibility in the direction parallel and
perpendicular to the magnetically stable axis, respectively.
The origin of has been attributed to the diamagnetic and
paramagnetic anisotropy.1,2) The degree of axis-alignment is
commonly described by an order-parameter hmi, which is the
Boltzmann average of a function ð3 cos2 1Þ=2 calculated
in terms of UðÞ as,
Z
hmi ¼ fð3 cos2 1Þ=2g expfUðÞ=kB Tg
Z
ð2Þ
sin d= expfUðÞ=kB Tg sin d:
Here hmi ¼ 0 and 1 correspond to completely random state
and to the completely ordered state of the grains, respectively. The order parameter has been commonly used to
describe the state of molecular orientation in the liquid
crystals as well as the state of magnetic alignment of the
micron-sized particles.1,2)
The measured hmi-B relationships of various inorganic
micro-crystals measured at room temperature are compiled in
Fig. 1, namely for kaolinite, talc5) and graphite.7) The
experimental hmi-B relationship was obtained from the light
intensity IðBi Þ transmitted through the sample-suspension at
an field intensity Bi .1,3,5) The field intensity was increased
stepwise until IðBi Þ ¼ IðBi Þ Ið0Þ reached a constant value
IS , where the alignment was almost completed. The hmi
values are obtained from the relation hmi ¼ Ii =IS , which
was deduced theoretically18) and confirmed experimentally
by measuring the hmi-B relation for several materials which
had published values determined by bulk measurements.5–7) The N value which gave the best fit to the
experimental hmi-B data was adopted to describe the
theoretical curves by the use of eq. (2). The adopted N
value for each curve is interpreted as the average N value
of the particles dispersed in the suspension.5–7,19,20,22)
The state of full orientation corresponding to the state of
hmi ¼ 1 cannot be obtained exactly at a finite field intensity
according to eq. (2), although the alignment seem to be
completed according to Fig. 1 for graphite, talc and kaolinite.
It is necessary therefore to define a field intensity where the
ð3Þ
It is deduced from eqs. (1) and (2) that a BS value correspond
to a unique hmi-B curve. The equation shows that a BS value
is determined by the three parameters7) as described before,
namely T, and N.
The BS –N relationships were examined experimentally for
inorganic materials, such as kaolin,5) talc,5) lepidorite6) and
fluoro-phlogopite.6) The expected N dependence of BS
described in the equation was confirmed in the range of N ¼
1010 to 1012 mole. The BS –N relationships measured for
several different materials at a constant temperature fell on
different lines, which derived from the difference of values. The results revealed that the values of individual
materials determined the BS values as well.
The T dependence of BS expected in eq. (3) was examined
experimentally between room temperature and 180 K for
synthetic graphite grains, free of paramagnetic ions, by
Chihara et al.7) Ethanol was chosen as the dispersing medium
for the low temperature experiments. The BS -T relations
observed for the two samples, namely graphite-1 and -2,
followed the theoretically expected T dependence as described in Fig. 2. The differences of the BS values observed
between the two graphite samples were explained by the
difference of the average N values obtained for graphite-1
and -2. The measurement served as an experimental
examination of the Langevin theory on magnetic alignment,
which assumed that the alignment was controlled by the
Brownian thermal energy. It is deduced from the results of
the following section that the reduction of BS by lowering the
temperature of the suspension, observed in Fig. 2, is
applicable for many of the diamagnetic oxides.
0.06
0.05
log (Bs/K)
0
graphite-1
graphite-2
0.04
0.03
0.02
100
200
300
400
500
log (T/K)
Fig. 2 Temperature dependence of magnetic alignment process observed
for diamagnetic graphite grains.
2596
C. Uyeda, K. Tanaka, R. Takashima and M. Sakakibara
Table 1
ðÞ [109 emu/g]
Material
15)
Diamagnetic Anisotropy Measured for Various Inorganic Oxides Material
Magnetically stable axis
Locality of product
y
11 0:5
a-axis
synthesized
apophylite14) [a-c]
brucite13) [c-a]
3:8 0:1
2:6 0:2
a-axis
c-axis
Jalgon-India, Poona-India
Zimbabwe, Ural-Russia
corundum16) [c-a]
0:7 0:1
c-axis
synthesized
4:2 0:3
c-axis
Mugla-Turkey
Gerais-Brazil
ADP
[a-c]
16)
diaspore
[c-a]
gibbsite13) [c-a]
1:4 0:2
c-axis
graphite10) [c-a]
0:15 0:2
c-plane
synthesized
gypsum15) [1 -2 ]
9:6 0:2y
1 -axis
Chihuahua-Mexico
muscovite13) [c-a]
11 2
c-axis
Minas Gerais-Brazil
KDP15) [a-c]
orthoclase14) [3 -2 ]
8:3 0:3y
2:1 0:1
a-axis
3 -axis
synthesized
Udateha-Russia, Gifu-Japan
-quartz10) [a-c]
2:0 0:2
a-plane
synthesized
kaolinite19) [c-c? ]
12 6z
c-plane
Georgia-USA, SouthCarolina-USA
talc20) [c-c? ]
77 38z
c-axis
Kouchi-Japan, Nagasaki-Japan
2 1y
c-plan
synthesized
phlogopite6) [c-c? ]
#1) Symbols [*] show that ðÞDIA were obtained from high temperature limit of -T measurement on bulk samples; see Fig. 4(a). Symbols [y] denote
that the samples had high quality free of paramagnetic impurity ions, and ðÞDIA were obtained by measuring bulk samples at room temperature. ðÞDIA
of phlogopile was obtained from magnetic alignment experiment on micro-crystals.6) Symbols [z] show that ðÞDIA was estimated from high temperature
limit of -T relations obtained from magnetic alignment experiment on micro-crystals;.20,21)
Effect of Diamagnetic Anisotropy on Magnetic Alignment
The number of published ðÞDIA values on basic
inorganic oxides have increased recently as mentioned
before.13–15) The values are listed in Table 1. The origin of
the obtained ðÞDIA values were explained consistently by
assigning a constant amount of uni-axial diamagnetic
anisotropy to the individual bonding orbital composing the
material.12,14–16) The direction of the principle axis was
assumed to be identical to the bond direction in the model.
Accordingly, positive correlations were expected between
measured ðÞDIA (per moll) and the differences between
2 , 2 and 2 as described in the appendix. , and denoted the direction cosines of the bond directions.
As a first step, the correlations mentioned above were
examined separately on three types of basic chemical bonds,
namely the T-O bonds composing the tetrahedral [TO4 ]
units,14) the hydrogen bonds15) and the M-O bonds composing the octahedral [MO6 ] units.16) The correlation of the
hydrogen bonds were examined using the experimental
ðÞDIA data of gypsum, KDP, ADP and hexagonal ice.15)
The relationship for T-O bonds was examined using the data
of quartz, orthoclase and apophylite.14) Finally the model was
examined for the M-O bonds using the data of corundum,
gibbsite, brucite13) and diaspore.16) Detailed procedures of the
calculation are given in the appendix.
Clear correlations were obtained for the three types of
chemical bonds as described in Fig. 3. Diamagnetic anisotropy deriving from a single chemical-bond ðÞBD were
estimated from the regression lines of the correlations to be
2:2, 0:63 and 0:19 [106 emu/mol] for the T-O bond,
hydrogen bond and the M-O bond, respectively; equations for
the regression lines are given in eqs. (A·2a)-(A·2c). It is noted
that the three types of chemical bonds are the major
categories of bonding orbital that compose the inorganic
oxides of light elements.16) Accordingly most of the
Tetrahedral unit
measured ∆χ (x10-6 emu/mol)
3.
2.5
2.0
1.5
Hydrogen bond
1.0
0.5
Octahedral unit
0.5
0
-0.5
-1.0
-1.5
-2.0
calculated Σ∆
Fig. 3 Correlation between Diamagnetic Anisotropy and Bond Direction.16)
diamagnetic oxide may posses a finite ðÞDIA value to
cause magnetic alignment, even if their concentrations of
electron spins are negligibly low.
It was expected in eq. (3) that BS was proportional to the
square root of T for diamagnetic material with high purity;
the relation was confirmed experimentally as shown in Fig. 2.
This BS -T relation is expected to occur for the abovementioned oxides as well. BS value at T ¼ 10 K for a
diamagnetic particle is expected to be less than one fifth of
the intensity required for alignment at room temperature,
Characteristics of Paramagnetic and Diamagnetic Anisotropy which Induce Magnetic Alignment of Micron-Sized Non-Ferromagnetic Particles 2597
since BS ðT ¼ 10 KÞ=BS ðT ¼ 300 KÞ ¼ ð10=300Þ1=2 according to eq. (3).
Double oxides, such as forsterite, kaolinite, muscovite and
phlogopite listed in Table 1, are composed of more than one
kind of chemical bonds. Hence, as a second step, the
measured ðÞDIA values of these oxides should be examined
to be consistent with the sum of ðÞBD values of different
chemical bonds. Numerical analysis is now in progress to
examine the efficiency of the above-mentioned model on
double oxides.
4.
Effect of Paramagnetic Anisotropy on Magnetic
Alignment.
30
(a) muscovite
20
10
0
1
2
3
-10
0.0
1.0
2.0
3.0
4.0
5.0
1.0
6.0
(x10-3)
T-1/103K-1
Magnetic Anistropy ∆ µ /Hm2Kg-1
Magnetic Anisotropy 10-9 emu/g
The -T relations of three muscovite crystals, which
have been measured to determine the ðÞDIA value, are
shown in Fig. 4(a).13) The ðÞDIA values were determined
from the high temperature limits of the T- relations. The
high temperature limits of were obtained form hmi-B
measurements as well performed for two different kaolinite
samples, as shown in Fig. 4(b).19) Kaolinite is a classical
ceramic-material that is difficult to synthesize; it usually
contain finite amount of paramagnetic impurity ions. Liquid
ethanol was adopted as the suspending medium, and the
measurement was performed throughout its liquid phase
temperature-region between T ¼ 200 K and 340 K. Linear
-1=T relations are obtained for the two samples, confirming the Curie-temperature dependence of . The quantita-
(b) Kaolinite
1
2
0.5
0.0
1.0
2.0
3.0
4.0
5.0
6.0
T-1/103K-1
Fig. 4
Temperature dependences of paramagnetic anisotropy.
tive amount of ðÞDIA and ðÞPARA components are
obtained from Fig. 4 for the measured materials. The
ðÞPARA components, following the Curie-law, were considerably large compared to the ðÞDIA components in the
low temperature region for the three muscovite samples as
well as for kaolinite-1. This tendency was seen in many of the
-T measurements on bulk oxide crystals, in the course of
determining ðÞDIA values from the high temperature
limits.13) The large ðÞPARA value with respect to the
ðÞDIA value indicate that the field intensity required to
achieve magnetic alignment of micron-sized particles may be
reduced considerably, when some amount of paramagnetic
impurity ions were contained in the crystals.
It is expected that the Bs value becomes nearly proportional to temperature in the low temperature region according
to eq. (3), since ðÞDIA becomes small compared to
ðÞPARA . Accordingly the reduction rate of Bs for the
paramagnetic grains with the decrease of temperature
becomes considerably large compared to that of pure
diamagnetic grains. The T dependence of Bs is expected to
occur for many of the non-ferromagnetic oxides, since the
ðÞPARA components were actually observed to be considerably large compared to the ðÞDIA values in many of the
measured oxides as mentioned above.
The efficiency of magnetic alignment at room temperature
can be enhanced when the contaminated paramagnetic ions is
increased in a diamagnetic insulator. It is seen in Fig. 4(b)
that the value of kaolinite-1, having a larger paramagnetic susceptibility (T¼300 K ¼ 190 106 emu/mol)
compared to kaolinite-2 (T¼300 K ¼ 72 106 emu/mol),
is always larger compared to the value of kaolinite-2. The
positive correlation between and seen for kaolinite was
observed in many of the oxides in the course of the bulk T measurements mentioned above. It is concluded that
doping a finite amount of paramagnetic ions during material
processing can enhance the practical efficiency of magnetic
alignment of non-ferromagnetic materials in many cases.
Paramagnetic anisotropy of muscovite and kaolinite are
considered to derive mainly from Fe2þ ions, since iron had
the highest concentration (order of several mass%) among
the paramagnetic impurity ions for the two materials.19,20)
The amount of ðÞPARA deriving from single Fe2þ ion can
be evaluated from anisotropy of g-value, gð¼ g k g ?Þ,
obtained by ESR measurements.21) g can be as large as 0.1
for Fe2þ ions occupying the octahedral [MO6 ] sites of the
sheet-silicates. It is noted that g-values are more or less
isotropic for Fe3þ ions occupying the above-mentioned sites.
Exact Fe2þ /Fe3þ ratios of the sample are therefore required
to evaluate the origin of ðÞPARA values in terms of abovementioned ESR analysis.
It is deduced from eq. (3) that Bs is reduced to 65 mT at
T ¼ 100 K for sphere-shaped grains of 1 mm in diameter
assumed for muscovite-1, according to the -T relation in
Fig. 4(a). The micron-sized insulators, however, do not
disperse effectively in cryogenic liquid such as N2 or Ar. A
medium that can disperse micron-sized crystals at low
temperature is the He gas. Magnetic alignment of micronsized graphite grains dispersed in a He gas medium at room
temperature was observed using an apparatus developed for
this purpose.22) Modifications of the apparatus are now
2598
C. Uyeda, K. Tanaka, R. Takashima and M. Sakakibara
carried out to examine the reduction of Bs value at low
temperature conditions,23) which has been proposed for
diamagnetic and paramagnetic materials in the present paper.
Acknowledgements
This work was partially supported by the Grant in Aid for
Scientific Research from the Ministry of Education, Science,
Sports and Culture, Japan (grant no. 14350008), by the Grant
in Aid for Scientific Research on Priority Area ‘‘Innovative
utilization of strong magnetic fields’’ (Area 767, No
15085206) and also by the Ground Research Foundation of
the Japan Space Forum.
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Appendix
Diamagnetic anisotropy are considered to originate from
spatial anisotropy of the electrons localized in a substance in
general; the spatial anisotropy of electrons in the inorganic
oxides are considerably small compared to those of the
organic molecular-crystals. It is noted that the diamagnetic
susceptibility of inorganic insulator is approximately equivalent to the simple summation of the susceptibility assigned
to the individual orbital consisting the material, according to
the advanced Pascal’s law.10) This treatment on diamagnetic
susceptibility can be expressed by a 3-demensinal -tensor of
a material, assuming that each orbital posses an uni-axial
anisotropy with constant value BO ¼ BO k BO ?;
BO k and BO ? are the susceptibilities parallel and
perpendicular to the bond direction.0,0,13,16) The field-induced
free energy UðBÞ of an orbital is described as,
UðBÞ ¼ ð1=2ÞB2 fBO ? þBO ða2 2 þ b2 2 þ c2 2 Þg:
ðA:1Þ
The direction cosines of B is defined as (a; b; c), whereas
(; ; ) denote the direction cosines for the bond direction;
magnetic principle axes are identical to the x-, y- and zcoordinates in the above vector components. , and are
calculated directly from atomic position data for various
materials. The ðÞDIA values (per moll) between x-y, y-z
and z-x axes of a material are expected to be proportional to
2 2 , 2 2 and 2 2 , respectively.
The relationships between experimental and calculated
anisotropy were obtained separately for the three chemical
bonds. The results are compiled in Fig. 3; solid circles, solid
squares and open symbols show the relationships for
materials composed of the T-O bonds, the M-O bonds and
the hydrogen bonds, respectively. Clear correlations, as
expected in the above-mentioned model, are seen for the
three types of chemical bonds. Regression lines for the three
types of bonds are calculated as,
O-H bond : ðÞDIA ¼ 2:2ðÞ þ 0:06½106 emu/mol ðf ¼ 0:99Þ;
ðA:2aÞ
T-O bond : ðÞDIA ¼ 0:63ðÞ þ 0:23½106 emu/mol
ðf ¼ 0:89Þ;
ðA:2bÞ
M-O bond : ðÞDIA ¼ 0:19ðÞ þ 0:03½106 emu/mol ðf ¼ 0:93Þ;
ðA:2cÞ
where the correlation factor is described as f . denote
the differences between 2 , 2 and 2 described in the
above equations. The gradient of the above regression line
correspond to the ðÞDIA value of a single bond. The large
differences seen between the obtained BO values of the
chemical bond probably derive from the differences of spatial
density distribution of each chemical bonds. These BO
values may serve as a basic data to construct a general theory
on diamagnetic anisotropy of inorganic insulators.