Biomedical Engineering Reference
In-Depth Information
16.4.2.3
Thermal-Activation Effect on the Moment and Coercivity
The S-W model does not cover thermally activated behavior. In fact, the magnetic
moment of single-domain ferromagnetic particles will decay towards thermal
equilibrium under the effect of thermal energy. For a single-domain ferromagnetic
particles with uniaxial anisotropy at zero applied fi eld and a certain temperature
KV
kT
u
B
(
T
), there are two minimum energy states separated by a energy barrier:
E
=
,
B
where
k
B
is Boltzmann's constant. When a magnetic fi eld
H
is applied to the
nanoparticles,
E
B
can be approximated by the following expression:
n
KV
kT
H
H
⎛
⎝
⎞
⎠
u
B
E
=
1
−
(16.5)
B
0
with
H KM
u
=
2
(16.6)
0
S
The fi eld
H
0
is defi ned simply as the fi eld at which the energy barrier is zero.
The external applied fi eld will decrease the energy barrier until switching occurs,
and thus
H
0
is also called the “switching fi eld ” ,
H
SW
. Therefore,
x
can be expressed
by:
3
2
(
)
−
2
3
2
3
x
=
sin
θ
+
cos
θ
(16.7)
Preiffer has shown that the S-W model yields the approximation for
n
: [129]
n
=+
086
.
114
.
x
(16.8)
When
H
is along the anisotropy axis of the particles,
x
= 1 and
n
= 2. For an
assembly of noninteraction particles, the anisotropy axis is 2-D randomly oriented,
which corresponds to
= 30 - 35 ° ;
n
= 1.43. Victora has reported the theoretical
calculation that, with the fi eld not aligned with the anisotropy axis,
n
would be
expected to be 1.5 for every general anisotropy, even including interaction [130].
Generally, the exponent
n
= 1.5 is used to evaluate thermal fl uctuation for the
nanoparticle with
H
not aligned with its anisotropy axis [131].
The probability of crossing the energy barrier per unit time can be expressed
by:
θ
1
E
kT
⎛
⎝
⎞
⎠
B
B
=
f
exp
−
(16.9)
0
τ
where
is known as the “characteristic relaxation time”. The moment
M
(
t
) of the
noninteracting particles assembly then decays exponentially with time from the
initial saturated moment
M
0
as:
τ
( )
−
t
⎡
⎢
⎤
⎥
Mt
()=
M
0
2
exp
−
1
(16.10)
τ
