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The effect of anisotropy in the
Dimer Model on Ferrofluids in
One-Dimension.
Dima Al-Safadi
Advisor: Dr. Abdalla Obeidat
Co-Advisor: Prof. Nabil Ayoub
Contents
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Introduction
Theory
Randomized easy axis
Fixed easy axis
Special cases
Discussion and Conclusion
Introduction
Magnetic anisotropy
In the absence of the applied magnetic field, the
magnetic moments of the particles tends to align
themselves to the direction which makes the
magnetoststic energy minimum, this direction is
called the easy direction, briefly, magnetic
anisotropy shows how the magnetic properties
depend on the direction of measuring the
magnetization. There are several types of anisotropy
and the most common one is the magnetocrystalline
anisotropy.
Magnetocrystalline anisotropy
(crystal anisotropy)
When single domain fine particles magnetized
to saturation, the magnetization has an easy axis
along which is prefer to stay. In this case, the total
internal energy is minimum. Rotation of the
magnetization vector away from an easy axis is
possible only by applying an external magnetic
field. Thus, the magnetic energy is directiondependent, and this kind of energy is called
magnetic anisotropy energy. It is also called
magnetocrystalline anisotropy because it has the
same symmetry as the crystal structure of the
particle material.
one of the simplest expression of the
magnetic anisotropy energy that is uniaxial
in symmetry is
Ea  K V sin 
2
Ê



What is the ferrofluids?
Ferrofluid is a stable colloidal suspension consists of
single-domain fine magnetic particles (ferromagnetic or
ferrimagnetic particles), of diameter (30-150) in a carrier
liquid (such as water or kerosene). To prevent the
agglomeration and enhance the stability of the suspension,
a nonmagnetic surfactant layer (such as polymers) covers
particles. Recently, the publications concerned with
ferrofluids have increased because these materials have a
wide uses in industry, medicine and agriculture
Randomize easy axis on ferrofluid
dimer model in one-dimension
We will study the effect of the magnetic
anisotropy and the effect of the dipole-dipole
interaction on the magnetic properties of a dilute
ferrofluid consists of a single domain magnetic fine
particles.
The easy axis of each particle ( Ê )directed

randomly and the magnetic moment (  ) is freely
to rotate in three dimensions.
Dimer model
To simplify our calculations, we will use a
simple theoretical model called (Dimer model)
in this model, the uniaxial single domain
magnetic fine particle interacts with only one
other adjacent particle thus, our assembly
consists of ( N 2 ) independent non-interacting
systems move in one dimension. Since each
system composed of two interacting particles,
we call it a dimer.
Ê 
Ê 

x
For one system we can write


   cos iˆ  sin  cos ˆj  sin  sin  kˆ
Eˆ  sin  cosiˆ  sin  sin ˆj  cos kˆ

Theoretical back ground
The partition function for one system is
Z  e
 ET
kT
d
ET  Eint  E0  Ea
Where
and

  
.  3( .r )(  .r )
Eint.  3 
5
r
r
   
E0  .H   .H
Ea  KV sin 2 
Therefore
Z  e
 ( Eint  E0  Ea )
kT
d
To calculate This integral we assume the
following:
1-The interparticle interaction energy is small
compared with thermal agitation energy
2-The anisotropy energy is small compared
with the thermal energy
3-The applied magnetic field is very small.
For ( N 2) system the total partition function is
N 2
given by

Z
ZT 
( N 2)!
The magnetization is defiand as
 ln ZT
  kT

And the initial susceptibility is given by
 M 
  lim 

H  0 H


Field parallel to the assembly
The following expression have been
calculated for the partition function
Z  C0i02  C1i12  C2i02  C3 F1 x   C4 F2 x   C5i12
Where
2 KV
4
kT
C0  256 e
( z0  zi )
C1  256
4
2
kT
e
 2 KV
kT
 1
1 
 2  2 
 zi z 0 
KV 2 KV kT
C 2  512
e
( z0  zi )
3kT
4
C3  16
4
4
2
10k T
2
e
 2 KV
kT
 1
1 
 5  5 
z0 
 zi
2
2
K V 2 KV kT
z 0  z i 
C 4  32
e
2 2
2k T
4
C5  1024
 KV
2
4
2
6k T
2
e
 2 KV


1
1
kT 


 z2 z2 
0 
 i
And
F1 x 
256
288 2 

2
64i0  x i0i1  x 2 i1 


F2 x 
[
18
224 2
i0 i1 
i0 ]
5x
45
The magnetization M is given by
C5
C1

C


C

C
33
.
3

C
6
.
3

2
3
4
Nx  0 3
3
M

 66
298
3 
C0  C 2  C3 (
)  C4

9
45





And the initial susceptibility is calculated to
be
N 2
 
3k
C1T
T 
3C 0
However, the general form of Curie-Weiss law is:
C

T  T0
Therefore our result obeys Curie-Weiss law with
and
N
C
3k
2
 z 0  z i 
T0 
2 2
3k z 0 zi 
2
Magnetic field perpendicular to
the assembly
Thank you