Biomedical Engineering Reference
In-Depth Information
An imperfect
H
/
T
superposition can result from a broad distribution of particles
size, changes in the spontaneous magnetization of the particles as a function of
temperature, or anisotropy effects [132]. The magnetization of superparamagnetic
particles may be used to determine the mean particle size and width of the size
distribution when a distribution form of the particles was given [133, 134].
The temperature - dependent magnetization,
M
(
T
), of a particle with given size
exhibits a peak in the zero-fi eld - cooled ( ZFC ) magnetization at
T
B
, but exhibits a
gradual decrease in the fi eld-cooled (FC) magnetization. In particles assembly, the
distribution of particle size may also cause a distribution of the blocking tempera-
ture [135-137]. Other than the ZFC and FC curve characterizations of superpara-
magnetism, some direct investigations of superparamagnetism have also been
proposed [94] . Woods
et al.
suggested the direct probing of superparamagnetism
in fi lms of self-assembled cobalt nanoparticles by measuring the spontaneous
magnetic noise arising from these fi lms as a function of temperature [94]. Nanopar-
ticle spin fl ips, induced by thermal energy, are directly sensed by a micro-SQUID
technique, providing statistical information on the magnetic properties of millions
of particles in the array. In this way, not only can the average anisotropy energy
and width be determined, but the entire magnetic anisotropy energy distribution
can also be extracted. This technique is a complete “magnetic fi ngerprint ” .
For superparamagnetic particles without interparticle interaction and having a
simple bistable state, the magnetic noise power measured at temperature
T
, cyclic
frequency
ω
, and distance
d
away, using a micro-SQUID method, is [138, 139]:
(
)
1+
2
μ
MV
d
τ
ωτ
(
) =
0
ST
ω
,
(16.15)
B
4
π
3
22
where
M
is the particle magnetization,
V
is the particle volume, and
τ
is the average
cyclic fl ipping time for the moment.
For a distribution
D
(
U
) of particle anisotropy energies, one fi nds for the noise
power (approximating with use of an average prefactor to the Lorentzian in Equa-
tion 16.15 ):
τ
ωτ
e
UkT
UkT
B
∞
(
) ∝
∫
0
2
ST
ω
,
DU dU
()
(16.16)
B
1
+
0
22
e
0
B
Based on Equation 16.16, the noise power is a function of cyclic frequency and
temperature. It was found from a plot of noise power as a function of frequency
from a fi lm of 5 nm Co nanoparticles at various temperatures, that the noise
decreases monotonically in the frequency window for each particular temperature,
having an approximate form of
S
B
f
− α
.
Based on Equation 16.16, the anisotropy energy distribution of Co nanoparticles
can be found from the noise power according to:
∼
2
ω
DkT
(
−
log
(
ωτ
)
)
∝
kT
ST
(
ω
,
)
(16.17)
B
0
B
π
B
