Effect of distributed particle magnetic moments on the magnetization of NiO nanoparticles

Article abstract of DOI:10.1016/j.ssc.2012.03.003

Magnetization of NiO nanoparticles, at different temperatures, is measured and analyzed taking into account a distribution in particle magnetic moment. We find that disregarding this distribution in the analysis is the reason for the many anomalous observations reported on this system in the literature.

Full text of DOI:10.1016/j.ssc.2012.03.003

Solid State Communications 152 (2012) 1080–1083
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Solid State Communications
journal homepage: www.elsevier.com/locate/ssc
Effect of distributed particle magnetic moments on the magnetization of
NiO nanoparticles
S.D. Tiwari a,n, K.P. Rajeev b
a School of Physics and Materials Science, Thapar University, Patiala 147004, India
b Department of Physics, Indian Institute of Technology, Kanpur 208016, India
a r t i c l e i n f o
a b s t r a c t
Article history:
Magnetization of NiO nanoparticles, at different temperatures, is measured and analyzed taking into
account a distribution in particle magnetic moment. We find that disregarding this distribution in the
analysis is the reason for the many anomalous observations reported on this system in the literature.
& 2012 Elsevier Ltd. All rights reserved.
Received 23 August 2011
Received in revised form
27 February 2012
Accepted 1 March 2012
by D.D. Sarma
Available online 29 March 2012
Keywords:
A. Nanoparticle
D. Superparamagnetism
E. Magnetization
E. Susceptibility
1. Introduction
behavior. We also showed that the low temperature behavior
of NiO nanoparticles is not superparamagnetic contrary to pop-
Studies on particles of a few nanometers in size have been
attracting quite a lot of interest of late. For a physicist the
attraction is mainly due to the emergent properties that a particle
or a collection of them shows when the particle size is made very
small. According to Ne´el tiny particles of an antiferromagnetic
material should exhibit magnetic properties such as superpar-
amagnetism and weak ferromagnetism [1]. If the surface to
volume ratio, which varies as the reciprocal of particle size, of
an antiferromagnetic particle is made sufficiently large then it can
have a nonzero net magnetic moment because of an imperfect
cancellation of elementary moments pointing in different direc-
tions near the surface of the particle. Recently nanoparticles of
antiferromagnetic materials have gained quite a lot of attention
mainly because they show some surprising and unusual behavior
unobservable in ferro or ferrimagnetic nanoparticles [2,3]. Among
different antiferromagnetic nanoparticles, NiO is a comparatively
more interesting and well studied system [2,4–6]. Long back,
Richardson and Milligan [4] reported magnetization measure-
ments on NiO nanoparticles of different sizes and this has been
followed by many more reports [2,5,6], mostly by other workers.
We also reported some work on NiO nanoparticles [7,8] where we
found that, at low temperatures, this system shows spin glass
ular belief and expectation. After our work some other workers
have also reached similar conclusions on NiO nanoparticles
independently [9].
The magnetization of antiferromagnetic nanoparticles is
expected to be described by
a modified Langevin function
[6,10]. However, fitting the magnetization data of bare antiferro-
magnetic NiO nanoparticles as a function of magnetic field to the
modified Langevin function results in unphysical fit parameters.
For instance, the estimated particle magnetic moment turns out
to be about 2000 mB for 5.3 nm particles [6]. Such values for the
particle magnetic moments are much larger than the expected
value of about a few hundred Bohr magnetons from uncompen-
sated spins on the surface of particles. Another method for the
estimation of particle magnetic moment of NiO nanoparticles has
also been used [2]. Here the value of saturation magnetization at
low temperature is divided by the estimated number of particles
to get the particle magnetic moment. However, the moment
estimated by this method has also been found to be larger than
the expected value. A multisublattice model has been proposed to
explain this large value of particle magnetic moment for NiO
nanoparticles though there is no experimental support for this
model yet [2].
The magnetization of nanoparticles of magnetic materials is
expected to be only a function of the applied magnetic field B and
temperature T and should scale with ðB=TÞ above the bifurcation
temperature (Tbf ) between low field cooled and zero field cooled
n Corresponding author. Tel.: þ91 175 2393116; fax: þ91 175 2364498.
E-mail address: sdtiwari@thapar.edu (S.D. Tiwari).
0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ssc.2012.03.003

S.D. Tiwari, K.P. Rajeev / Solid State Communications 152 (2012) 1080–1083
1081
magnetization. The magnetization of NiO nanoparticles is not
found to show this scaling [6]. These observations motivated us to
revisit the antiferromagnetic NiO nanoparticles system once
again. This work is an attempt to find the reason for getting
unphysical numbers for the particle magnetic moment of NiO
nanoparticles when the traditional methods are used to analyze
the magnetization data above Tbf . The value of the bifurcation
temperature Tbf for the present system is about 295 K in 100 G
applied magnetic field [8].
superparamagnetic at sufficiently high temperature (T 4Tbf ) [1].
Magnetization M of a superparamagnetic material as a function of
magnetic field B and temperature T is described by [12]
M ¼ M0LðxÞ,
ð1Þ
where LðxÞ ¼ ½cothðxÞꢀð1=xÞꢁ is the Langevin function and
x ¼ mpB=kBT. Here M0 is the saturation magnetization, mp is the
particle magnetic moment and kB is the Boltzmann constant.
However, magnetization of small particles of antiferromagnetic
materials, even though they are superparamagnetic cannot be
described by Eq. (1). Rather, they are well described by an altered
form known as the modified Langevin function [10,13], which
contains an extra linear term in B i.e.
Here we present the magnetization measurements on 5 nm
bare NiO particles as a function of applied magnetic field at
´
sufficiently high temperatures, but well below the Neel tempera-
ture (TN) of the system which is known to be about 523 K [11].
We analyzed the data using the modified Langevin function
without considering any distribution in particle magnetic
moment and then repeated the analysis taking into account a
distribution in the particle moment. We were pleasantly sur-
prised by the results as it turned out that we can account for the
anomalous observations reported on this system by the earlier
workers.
M ¼ M0LðxÞþ
waB,
ð2Þ
where wa is the susceptibility of randomly oriented antiferromag-
netic particle cores. We fitted the data shown in Fig. 1 to the
above equation and the resulting fits are shown as solid lines in
the figure. Values of fit parameters M0, mp and wa obtained are
presented in Table 1.
From Fig. 1 we see that the fits to Eq. (2) more or less pass
through the measured data points which means the fit quality is
quite good. A statistical measure of the goodness of a fit is the
coefficient of determination R2. The closer R2 is to unity the better
the fit. Values of the coefficient of determination R2 for the fits are
shown in Table 1 along with the fit parameters. The R2 values are
greater than 0.999 in all cases and this once again confirms the
good quality of the fits.
2. Experimental details
NiO nanoparticles were prepared by a sol-gel method by
reacting in aqueous solution nickel nitrate and sodium hydroxide
at pH¼12 at room temperature as described elsewhere [4,5,6].
We used nickel (II) nitrate hexahydrate (99.999%), sodium hydro-
xide pellets (99.99%), both from Aldrich, and triple distilled water
to make nickel hydroxide. The sample of nickel oxide nanoparti-
cles was prepared by heating the nickel hydroxide at 523 K for
3 hours in flowing helium gas (99.995%). The sample was
characterized by x-ray diffraction and transmission electron
microscopy. The average crystallite size as well as the particle
size were found to be about 5 nm. The details of sample synthesis
and structural characterization have been reported in one of our
earlier works [8]. All the magnetic measurements were done
with a commercial SQUID magnetometer (Quantum Design,
MPMS XL5).
From Table 1 we see that the particle magnetic moment mp is
about two thousand Bohr magnetons and it decreases with
increasing temperature. Somewhat similar results have been
reported by others on NiO nanoparticles [6] as well as on NiO
nanorods [14]. However, there is a glaring inconsistency between
the numbers we get for M0 and mp. In Eq. (2) the fit parameters M0
and mp should be related as M0 ¼ Nmp where N is the number of
particles per unit mass of the sample. If we use this relation to
estimate the average particle magnetic moment mp it turns out to
be about 60 mB at 310 K which is very small compared to the
value of mp presented in Table 1. Alternatively if we use this
relation to estimate the value of N using the values of M0 and mp
from the table it turns out to be about 7.1 ꢂ 1016
g
ꢀ1. However,
1 g sample should contain about 2.2 ꢂ 1018 NiO particles of 5 nm
size. These observations invalidate the fit of experimental data to
Eq. (2). In the next paragraph we shall give another argument
against the numbers presented in Table 1.
3. Results and discussion
We measured the magnetization M of the 5 nm NiO particles
as a function of external applied magnetic field B at different
temperatures T (Tbf oT oTN). These measurements are shown in
Fig. 1. Small particles of magnetic materials are expected to be
´
Neel had discussed various ways by which a magnetic moment
can appear on an NiO particle due to incomplete compensation of
´
atomic moments in different sublattices [1,5]. According to Neel
the particle magnetic moment of an NiO nanoparticle can be
written as
mp ¼ pmAmB
:
ð3Þ
Here p is a number which depends on the size, crystal structure
and form or shape of the particle and mA is the magnetic moment
Table 1
Values of fit parameters M0, mp and wa to Eq. (2) and the corresponding values of
the goodness of fit parameter R2 for the 5 nm NiO nanoparticles at different
temperatures.
T (K)
M0 (emu/g)
mp
(
mB
)
wa (10ꢀ6 emu/g Oe)
R2
310
320
330
340
350
1.30
1.18
1.05
0.94
0.82
1967
1841
1825
1734
1621
34
32
31
29
27
0.9993
0.9994
0.9995
0.9996
0.9997
Fig. 1. (Color online) Magnetization as a function of applied magnetic field for
5 nm NiO particles at different temperatures. Solid lines show fits to Eq. (2). The
inset shows a magnified view of the data along with the fits at lower magnetic
fields which makes it clear that the fit quality is poor there.

1082
S.D. Tiwari, K.P. Rajeev / Solid State Communications 152 (2012) 1080–1083
of Ni2þ ion which is known to be 2 mB [2,15]. Neel proposed a
total of five different possibilities for the origin of magnetic
moment on an NiO particle of cubic shape. Out of the five, two
possibilities give the values of particle magnetic moment to be
zero corresponding to p¼0. The other three possible values of p
Table 2
´
Values of fit parameters N, s, n and wa to Eq. (5) and the values of R2 for the 5 nm
NiO particles at different temperatures. The mean magnetic moment of a particle
estimated from the fit parameters are also shown.
T (K) N (1017/g)
s
n ðmB
Þ
wa
mmean
ð
mB
Þ
R2
(10ꢀ6 emu/g Oe)
1=2
2=3
are n1io=n3s, n
and n
where nions is the number of Ni2þ ions in
ions
ions
the particle. Now, there can be many other possible set of values
of p corresponding to shapes such as spheres, tetrahedrons or,
more realistically, irregular shapes. But it seems safe to assume
310
320
330
340
350
8.7
7.3
6.5
6.9
6.8
1.34 116.5 21.4
1.30 130.3 21.4
1.27 136.8 20.9
1.28 118.9 19.8
1.26 114.4 18.9
287
305
308
271
254
0.999997
0.999998
0.999997
0.999998
0.999998
2=3
ions
that the upper limit of p will be about n
and the average will be
considerably less. Making use of the fact that the crystal structure
˚
of NiO is face centered cubic with lattice constant 4.176 A, the
value of nions for a 5 nm diameter spherical particle turns out to be
about 3592. Using the information presented above along with
Eq. (3) we get the particle magnetic moment to be about 31 mB
,
1=3
1=2
2=3
120 mB and 469 mB corresponding to p values of n , n
and n
ions
ions
ions
respectively. Thus according to Ne´el’s picture the maximum
possible value of particle magnetic moment for a 5 nm NiO
particle would be about 469 mB which is much smaller than the
values shown in Table 1.
While using Eq. (2) to fit the magnetization data we made a tacit
assumption that all the particles have the same magnetic moment.
But, this is not true. A sample of NiO nanoparticles has a distribution
of particle magnetic moments not only due to distribution in size
and shape but also due to the way in which the imbalance of spins
´
arises in the particle as pointed out by Neel [1,5]. We hazard the
guess that disregarding the distribution in particle moments is
perhaps the reason for the unphysical fit parameters obtained in
Table 1. Now we would like to carry out this fitting taking into
account a moment distribution. Making our path easier are pre-
cedents in this kind of analysis set by many workers in analyzing the
magnetization of different nanoparticle systems [16–18].
Fig. 2. (Color online) Magnetization as a function of applied magnetic field for
5 nm NiO particles at different temperatures. Solid lines show the fits of data to
Eq. (5). The inset shows the fits at low magnetic fields.
fit parameters thus obtained along with the values of coefficient
of determination R2 are shown in Table 2. We see in Table 2 that
the values of the coefficient of determination R2 are surprisingly
greater than 0.99999 in all the cases. These values of R2 are much
better than the values of the same shown in Table 1. This means
that the quality of the fits using Eq. (5) is much better than that
using Eq. (2). In Fig. 2 we show the new fits which clearly look
good. The inset clearly shows that the fits are very good for lower
values of applied magnetic fields also where the effect of
distribution in the particle magnetic moment on the magnetiza-
tion of the system is important as already discussed. We can now
conclude that the fits of the magnetization data to Eq. (5) are
much better than that to Eq. (2).
In a sample that has a distribution in particle magnetic
moments the low field magnetization is governed by the particles
with larger magnetic moments. The contribution of particles of
lower magnetic moments to the magnetization becomes impor-
tant only at higher applied fields where the high field forces the
moments to align with the field. The effect of a distribution in the
particle magnetic moment on the magnetization of the system
will show up in the Langevin or modified Langevin fit. To see this
dependence let us take a look at the comparison of the measured
magnetization to the modified Langevin fit at low fields in the
inset of Fig. 1. It is clear that the fits are not good at low fields
while from the main panel one can see that the situation is much
better at higher fields. This low field misfit is an indication of the
role a distribution in the particle magnetic moment plays on the
magnetization of a system. In this case the modified Langevin
function has clearly underestimated the contribution of the larger
moments.
From Table 2 we find that the value of N is now closer to the
expected value of about 2.2 ꢂ 1018
g
ꢀ1. Also the mean particle
pffiffiffiffiffiffi
2
magnetic moments, which are equal to n es , turns out to be
about a few hundred Bohr magnetons, rather than a few thousand
Bohr magnetons as found in the earlier fit, and are also well below
the maximum possible value of about 469mB according to the
Following Silva et al. we assume that in a sample of nanopar-
´
models proposed by Neel for the origin of magnetic moment in
ticles the distribution in particle magnetic moment,
described by a log normal distribution function of the form [17]
m, can be
NiO particles [1,5]. Fig. 3 shows variation of the mean particle
magnetic moment mmean and the susceptibility of antiferromag-
netic cores wa as a function of temperature. Histogram in Fig. 4
shows distribution of particle size in our sample. This histogram
has been taken form one of our works [8]. The value of particle
magnetic moment and so also the particle magnetic moment
distribution depends on the value of parameter p in Eq. (3). Three
curves in this figure show variation of particle magnetic moment
distribution corresponding to three different non zero values of p
for cubic shaped NiO particles, as already discussed. Somewhat
similar types of comparison between particle size distribution
and the particle magnetic moment distribution are also shown by
others for different nanoparticle systems [16]. Here it is worth
mentioning that the particle size distribution and the particle
h
ꢀ ꢁi
2
m
n
2s2
ln
1
pffiffiffiffiffiffi
fð
mÞ ¼
expꢀ
,
ð4Þ
ms
2
p
where n and s are parameters that characterize the distribution.
pffiffiffiffiffiffi
2
The mean particle magnetic moment mmean is equal to n es . Now,
Eq. (2) takes on the new form
Z
MðB,TÞ ¼ N 1 mLðxÞfð
mÞ d
m
þ
w
aB,
ð5Þ
0
where N is the total number of particles contributing to magne-
tization and L(x) is the Langevin function with x ¼ B=kBT. We
used this equation to fit the magnetization data of Fig. 1 and the
m

S.D. Tiwari, K.P. Rajeev / Solid State Communications 152 (2012) 1080–1083
1083
of Ni2þ moment makes only a minute shift in the particle
magnetic moment distribution shown in Fig. 4 towards the left.
4. Conclusion
In conclusion, we reported magnetization as a function of
applied magnetic field for 5 nm NiO particles at different tem-
peratures above the bifurcation temperature Tbf . Fitting the
magnetization data to the modified Langevin function without
considering any distribution in the particle magnetic moments
yields very large and unphysical values for the particle magnetic
moment. However, we got reasonable values for the particle
magnetic moment if the modified Langevin function is used to
fit the magnetization data taking into account a distribution in the
particle magnetic moment. The distribution in the particle mag-
netic moment arises from a distribution in particle size and form.
This work clearly shows that the non-consideration of a distribu-
tion in particle magnetic moment could be the reason for the
anomalously large values of magnetic moment of NiO nanopar-
ticles reported in the literature.
Fig. 3. (Color online) Variation of mean particle magnetic moment and suscept-
ibility as a function of temperature for 5 nm NiO particle. Values are taken from
Table 2. The lines joining the data points are guides to the eyes.
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´
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magnetic moment distribution are two different things. According
´
to Neel [1,5] there may be many particles not contributing to the
magnetization as already discussed. Also the exact value of
parameter p, on which the particle magnetic moment distribution
depends, depends on shape of the particle. It could be that for
these reasons the distributions shown in Fig. 4 do not exactly
overlap with the data. Also one must not forget that we have just
assumed that the moments are distributed in a log-normal
fashion which any way turns out to be a much better assumption
than the usual one: that the particle moments are distributed as a
delta function. While plotting the particle magnetic moment
distribution in Fig. 4 we have used the value of 2mB for the
magnetic moment of Ni2þ ion assuming the complete quenching
of orbital angular momentum in the system [2,15]. However, in
reality the orbital angular momentum in the NiO system is not
completely quenched and a value of 2.2 to 2:3mB has been
reported for the Ni2þ ions in the system [19]. Using this value
[19] W. Low, J.T. Suss, Phys. Rev. Lett. 15 (1965) 519.