162 8 Magnetic Nanomaterials, Superparamagnetism
temperatures. Other antiferromagnetic compounds that are possibly of interest in
this context are FeO and MnO.
8.3
Susceptibility of Superparamagnetic Materials
The susceptibility χ of a magnetic material is defined as χ =
∂
∂
=
M
H H 0
, the change
of the magnetization with the external magnetic field. Generally, the susceptibility
is given at low fields, in the vicinity of H = 0. Therefore, the magnetization, given
by the Langevin formula, Eq. (8.3a), is developed in a Taylor series around H = 0,
where the first element is used
M
nm
mH
kT
kT
mH
nm
kT
mH
mH
kT
kT
mH
H
H
→
→
=
−
≈
+
−
0
0
3
coth
=
nm H
kT
2
3
.
(8.10)
This leads for the susceptibility to
χ =
∂
∂
=
∝
=
→
M
H
nm
kT
nv
kT
vV
kT
H 0
2
2
3
specimen .
(8.11)
Equation (8.11) used the proportionality between the magnetization and the
volume of a particle (more exactly stated: the magnetically active volume). At constant volume of the specimen V specimen = nv, the susceptibility is larger if the volume
of the specimen is larger. Lastly, this says: if one wants to obtain material with
high susceptibility, one has to look for the largest possible particles.
There is a second property that must be taken into account; this is the relaxation
time, which determines the maximal frequency for application. As shown in Eq.
(8.5) the relaxation time τ N for Néel superparamanetism, τ 0 is set constant [1], an
assumption simplifying the considerations and not impairing the general tendency. To obtain more precise quantitative results, the refined theory of Aharoni
[2] should be applied.
τ
τ
N =
0
1
exp
.
K v
kT
(8.12)
From Eq. (8.12), one learns that the relaxation time increases exponentially with
increasing particle volume, which means with increasing particle size. In view of
the technical applications, selecting an appropriate particle size, one has to make
a compromise between high susceptibility, demanding large particles, and short
relaxation time, which is necessary for highfrequency applications, demanding
small particles. Certainly, with respect to the relaxation time, the magnetic anisotropy constant of the selected material is a further degree of freedom in design.
Typical values for the susceptibility as function of frequency. of different nanoparticulate ferrites are, in comparison with data for conventional ferrites, depicted
in Figure 8.16.
temperatures. Other antiferromagnetic compounds that are possibly of interest in
this context are FeO and MnO.
8.3
Susceptibility of Superparamagnetic Materials
The susceptibility χ of a magnetic material is defined as χ =
∂
∂
=
M
H H 0
, the change
of the magnetization with the external magnetic field. Generally, the susceptibility
is given at low fields, in the vicinity of H = 0. Therefore, the magnetization, given
by the Langevin formula, Eq. (8.3a), is developed in a Taylor series around H = 0,
where the first element is used
M
nm
mH
kT
kT
mH
nm
kT
mH
mH
kT
kT
mH
H
H
→
→
=
−
≈
+
−
0
0
3
coth
=
nm H
kT
2
3
.
(8.10)
This leads for the susceptibility to
χ =
∂
∂
=
∝
=
→
M
H
nm
kT
nv
kT
vV
kT
H 0
2
2
3
specimen .
(8.11)
Equation (8.11) used the proportionality between the magnetization and the
volume of a particle (more exactly stated: the magnetically active volume). At constant volume of the specimen V specimen = nv, the susceptibility is larger if the volume
of the specimen is larger. Lastly, this says: if one wants to obtain material with
high susceptibility, one has to look for the largest possible particles.
There is a second property that must be taken into account; this is the relaxation
time, which determines the maximal frequency for application. As shown in Eq.
(8.5) the relaxation time τ N for Néel superparamanetism, τ 0 is set constant [1], an
assumption simplifying the considerations and not impairing the general tendency. To obtain more precise quantitative results, the refined theory of Aharoni
[2] should be applied.
τ
τ
N =
0
1
exp
.
K v
kT
(8.12)
From Eq. (8.12), one learns that the relaxation time increases exponentially with
increasing particle volume, which means with increasing particle size. In view of
the technical applications, selecting an appropriate particle size, one has to make
a compromise between high susceptibility, demanding large particles, and short
relaxation time, which is necessary for highfrequency applications, demanding
small particles. Certainly, with respect to the relaxation time, the magnetic anisotropy constant of the selected material is a further degree of freedom in design.
Typical values for the susceptibility as function of frequency. of different nanoparticulate ferrites are, in comparison with data for conventional ferrites, depicted
in Figure 8.16.