8.2 Fundamentals of Superparamagnetism 153
The term “superparamagnetism” stems from the physical behavior of an ensemble
of such particles in a magnetic field. As for paramagnetic materials superparamagnetic materials follow Langevin’ formula for magnetization:
M nm
mH
kT
kT
mH
nmL
mH
kT
=
−
=
coth
.
(8.3a)
In this equation, m stands for the magnetic moment of one particle, which is, to
a first approximation, proportional to the volume of the particle, n the number of
particles, and H the applied magnetic field. For paramagnetic materials, the magnetic moment of one atom or molecule is very small, in most cases it is just one
Bohr magneton, μ B . The abbreviation L
mH
kT
is called the Langevin function. In
the case of superparamagnetic materials, the magnetic moment of one particle is
in the range of a few hundred, may be up to thousands of Bohr magnetons. The
product nm stands for the saturation magnetization. Analyzing Eq. (8.3a) in detail,
one realizes that is it possible to define a temperature compensated field H
H
T
*
=
(the term
m
k
H
T
is often called the “reduced field”.). Using this temperaturecompensated field, Eq. (8.3a) becomes
M nm
m
k
H
k
m H
=
−
coth
.
*
*
1
(8.3b)
As a consequence of Eq. (8.3b), one can check the magnetization curve for
superparamagnetism by plotting the data obtained at different temperatures
versus H
*
.
Box 8.1 Néel and Brownian Superparamagnetism
Both phenomena follow the Langevin formula, Eq. (8.3) and the magnetization
curve does not show any hysteresis, even when they are in their physical background, entirely different. The Brownian superparamagnetism is observed in
the case of free magnetic particles suspended in a liquid. Immersed in a liquid,
the particles are able to follow any change of direction of an external magnetic
field by rotation. The relaxation time, τ B , the time needed by the particle to
follow a change in the direction of the magnetic field, is given by [1]
τ
η
B =
3v
kT
.
(8.4)
The quantity η stands for the viscosity of the liquid and v for the volume of the
particle. The Brownian superparamagnetism is often called extrinsic in contrast
to Néel’s intrinsic superparamagnetism. In the Néel case, a spatially fixed particle is assumed. Therefore, if the external magnetic field changes its direction,
The term “superparamagnetism” stems from the physical behavior of an ensemble
of such particles in a magnetic field. As for paramagnetic materials superparamagnetic materials follow Langevin’ formula for magnetization:
M nm
mH
kT
kT
mH
nmL
mH
kT
=
−
=
coth
.
(8.3a)
In this equation, m stands for the magnetic moment of one particle, which is, to
a first approximation, proportional to the volume of the particle, n the number of
particles, and H the applied magnetic field. For paramagnetic materials, the magnetic moment of one atom or molecule is very small, in most cases it is just one
Bohr magneton, μ B . The abbreviation L
mH
kT
is called the Langevin function. In
the case of superparamagnetic materials, the magnetic moment of one particle is
in the range of a few hundred, may be up to thousands of Bohr magnetons. The
product nm stands for the saturation magnetization. Analyzing Eq. (8.3a) in detail,
one realizes that is it possible to define a temperature compensated field H
H
T
*
=
(the term
m
k
H
T
is often called the “reduced field”.). Using this temperaturecompensated field, Eq. (8.3a) becomes
M nm
m
k
H
k
m H
=
−
coth
.
*
*
1
(8.3b)
As a consequence of Eq. (8.3b), one can check the magnetization curve for
superparamagnetism by plotting the data obtained at different temperatures
versus H
*
.
Box 8.1 Néel and Brownian Superparamagnetism
Both phenomena follow the Langevin formula, Eq. (8.3) and the magnetization
curve does not show any hysteresis, even when they are in their physical background, entirely different. The Brownian superparamagnetism is observed in
the case of free magnetic particles suspended in a liquid. Immersed in a liquid,
the particles are able to follow any change of direction of an external magnetic
field by rotation. The relaxation time, τ B , the time needed by the particle to
follow a change in the direction of the magnetic field, is given by [1]
τ
η
B =
3v
kT
.
(8.4)
The quantity η stands for the viscosity of the liquid and v for the volume of the
particle. The Brownian superparamagnetism is often called extrinsic in contrast
to Néel’s intrinsic superparamagnetism. In the Néel case, a spatially fixed particle is assumed. Therefore, if the external magnetic field changes its direction,