Dipolar Interactions in Superparamagnetic
Nano-Granular Magnetic Systems
M. El-Hiloa, J. Al Saeib and
R. W Chantrellc
aDept.
of Physics, College of Science, University of Bahrain, P.O. Box
32038, Sakhir, Bahrain
bDept. of Physics, Imperial College London, London, SW7 2AZ
c Department of Physics, The University of York,York, YO10 5DD, UK
Introduction
 Dipolar interaction effects are complex many-body problems,
especially when the particles are poly-dispersed, located randomly,
and have their easy axes randomly oriented.
 Many phenomenological models have been suggested to account for
the interaction effects in fine particle systems.
 One of these models represents interactions via a mean statistical
field which can be added or subtracted from the applied field [1].
 Other models, the so-called T* model, in which a fictitious
temperature is introduced in the Langevin response of the
superparamagnetic system so that L(H/kT) is replaced by the
modified function L(H/k(T+T*)) [2].
[1] S. Shtrikmann and E.P. Wohlfarth, Phys. Lett. 85 A , 457 (1981).
[2] P. Allia, M. Coisson, M. Knobel, P. Tiberto and F. Vinai, Phys. Rev. B 60, 12 207 (1999).
Techniques such as Monte-Carlo simulations (MC) was found to be
very useful in examining the effects of dipolar interaction in random
systems.
Using MC simulations, Mao et al [3] have questioned the T* model
and concluded that this model does not adequately describe the
magnetization of interacting superparamagnetic assemblies.
The MC simulations of Chantrell et al [4] have shown that
magnetization curves are depressed with increasing particle
concentration which was attributed to the formation of flux closure.
In [4], the simulations of the initial susceptibility also predicted an
apparent negative ordering temperature. It has been noted that this kind
of order is not indicative of antiferomagnetic ordering.
[3] Z. Mao, D. Chen and Z. He, J. Magn. Magn. Mater. 320 (19), 2335 (2008).
[4] R.W. Chantrell, N. Walmsely, J. Gore and M. Maylin, Phys. Rev. B 63, 24410 (2000).
In a recent MC simulations study [5], the magnetization curves of a
nano-granular superparamgentic system are predicted to be always
depressed with increasing particle packing densities.
In [5], the distributions of dipolar interaction fields along the x, y and z
directions are predicted to be symmetric and Gaussian in form with an
average very close to zero.
In [5], the reduction in magnetization is attributed to the non-linear
response of the magnetization to the applied field, which weights the
negative interaction fields more strongly than the positive fields.
In this study, the effect of a Gaussian distribution of longitudinal
dipolar fields on the magnetization response of a nano-granular
superparmagnetic system in the limit of weak anisotropy is
examined, analytically and numerically.
[5] J. Al Saei, M. El-Hilo and R.W. Chantrell, Submitted to J. Appl. Phys. (2011)
The Model
For a distribution of longitudinal dipolar fields and in the limit of
weak anisotropy, the magnetization of identical superparamagnetic
particles is simply described by a Langevin function weighted by the
distribution of interaction fields, and is simply expressed as;

  ( H a  Hi ) 
M  Ha    L 
 f  H i  dH i
kBT



L(b) is the Langevin function
f  Hi   e
2
 H i2 / 2 Hi
/ 2 Hi is the Gaussian distribution of dipolar fields
with  Hi is the spread of dipolar fields.
 The distribution of longitudinal dipolar fields is assumed to remain
invariant with applied field. In real systems, f(Hi) is expected to
becomes narrower at high fields but it wont affect the results since;




 kBT

kBT
kBT
kBT
M  H a  / M s   1 
f
H
dH

1


H
f
H
dH

1





i
i
i
i
i


2


(
H

H
)

H

H
 Ha
a
i 
a
a

 
 
Analytical Approach
 At low fields, where b1, the Langevin function can be expanded and
then the magnetic response of the system can be expressed as:

  ( H a  H i )  3 ( H a  H i )3 2  5 ( H a  H i ) 5

M  Ha    



....
 f  H i  dH i
3
5
3kBT
45(kBT )
945(kBT )

 
 Since f(Hi) is symmetric, then the above equation can be simplified
to:
 Ha 
 2 Hi2
4

30 4 Hi
 3 H a3
M (Ha ) 

 .... 
1 
2
4
3
3kBT  5(kBT ) 315(kBT )
 45(kBT )
2
 20 2 Hi

1


....

  ....
2
 21(k BT )

 Thus every term in the Langevin expansion is depressed as a result of
a symmetric Gaussian distribution of longitudinal dipolar fields.
 The reduction in magnetization increases when the standard deviation
of dipolar longitudinal fields (Hi) is increased.
Thus, this reduction in the magnetization of the system cannot be used
to infer that the local interaction field is negative, since both negative
and positive interaction fields are equally probable.
When the magnetic response of the system is linear with applied field
i.e. M(Ha)(Ha+Hi), then a symmetric distribution of interaction fields
can have no effect on the magnetization because the weighted average
of the interaction field <Hi>=0.
Thus the reduction in magnetization for the system examined is
attributed to the non-linear response of the magnetization to the applied
field, which weights the negative interaction fields more strongly than
the positive fields.
The Initial Susceptibility
d (M / M s )
i 
dH a
H
4

30 4 Hi


 ....
1 
2
4
3k BT  5(k BT ) 315(k BT )

 
a 0
 2 Hi2
In the case where the standard deviation of dipolar fields is small i.e.
Hi<<kBT/, the third term in the above equation becomes negligible,
hence


C
i 


1
2 2
2 2

  Hi  T  T0 (T )


  Hi
3k T 
3k BT 1 
 ..
2
 5(k BT )

B


5k B2T 
Over a certain small range of temperatures, T0(T) varies slowly with
temperature, then
C
C
i 

T  T0 (T ) T   T0 
The predicted negative average ordering temperature is an
apparent measure of negative ordering in the system, since
positive and negative fields are equally probable.
Numerical Results
Reduced Magnetization (M/Ms)
1.0
0.8
0.6
Dm=6 nm
Hi=0 Langevin}
Hi=300 Oe
Hi=600 Oe
Hi=900 Oe
0.4
0.2
0
0
1000
2000
3000
The reduction in
magnetization is due
to the non-linear
response of the
magnetization to the
applied field, which
weights the negative
interaction
fields
more strongly than
the positive fields.
4000
Applied Field (Ha)
The calculated magnetization curves for a symmetric Gaussian distribution function of
longitudinal dipolar fields f(Hi) at different values of the standard deviation Hi=300,
600 and 900 Oe. The solid line represents the Langevin function, the case where
Hi0.
-1
Inverse Reduced Initial Susceptibility i (Oe)
-1
Numerical Results
900
when Hi=30 Oe,
i.e very small
compared to the
range of applied
fields for which
the magnetization
is still linear, the
variation of i-1 vs.
T extrapolates to
almost
zero
ordering
temperature
Hi=30 Oe
Hi=100 Oe
Hi=200 Oe
600
300
0
0
100
200
300
Temperature T(K)
The inverse of the calculated reduced initial susceptibility using a Gaussian distribution
function of longitudinal dipolar fields f(Hi) at different values of the standard deviation
Hi=30, 100 and 200 Oe.
Conclusions
The results show that interaction effects cannot be deduced
from changes in the average magnetization of the system.
For a symmetric distribution of longitudinal dipolar fields,
the magnetization of a superparamagnetic system is predicted
to be always depressed as the spread of dipolar fields is
increased.
The analytical analysis shows that this effect arises from the
non-linear response of magnetization, which weights
negative interaction fields more strongly than the positive
fields.
Thus the idea of describing dipolar interactions in terms of a
mean field that could be added or subtracted from the applied
field in random systems is not justified.
Conclusions
Due to the non-linear effect of magnetization, the
temperature variation of the initial susceptibility predicts an
ordering temperature that varies with temperature.
The predicted negative ordering temperature cannot be
considered as indicative of anti-ferromagnetic order.
Thus the idea of using an effective temperature (T*) to
represent interaction effects does not adequately describe the
magnetization of interacting superparamagnetic random
systems.