will lead to reduced susceptibility since, in a given volume, the content of active
material is reduced.
When considering superparamagnetism, distinction must be made between two
different definitions: (i) an older definition, which stems from Elmore [13] and is based
on the Brownian motion of magnetic particles in a liquid, and (ii) a younger definition,
as reported by N eel [11]. In the Brownian case, the particles are rotating in a liquid
when the direction of the magnetic field is changed. Based on these mechanisms, the
Brownian case is often referred to as “extrinsic” superparamagnetism, in contrast to
the “intrinsic” superparamagnetism in N eel’s case, which is the one discussed in the
preceding part of this chapter. The relaxation time (the time needed by the particle to
follow a change in the direction of the magnetic field) for the Brownian superparamagnetism is t B ¼ 3vg=kT, where g represents the viscosity of the liquid carrier
(the definitions of the other letters are outlined above). In the case of N eel’s superparamagnetism, the relaxation time is calculated using Eq. (8.11) [14]. The N eel case
will be discussed in detail later in this chapter. Both cases have in common that each
particle consists of a single domain and, provided that the measuring time constant is
adequate, hysteresis is not observed. Additionally, in both cases, the magnetization
curves follow Langevin’s law. When comparing the relaxation times for both cases, for
10-nm particles, in the Brownian case one finds values in the region of microseconds,
whereas in N eel’s case the relaxation time is about 1 ns. For magnetic particle with
sizes of approximately 1 mm, suspended in a liquid, the Brownian relaxation times are
close to 1 s. The relaxation time rules the ability of the particles to follow a change of an
external magnetic field with frequency f. In the case of t < 1/f, the particle is able to
follow the external frequency, and therefore remanence and coercivity are zero. In the
case of t > 1/f, hysteresis is observed.
In spatially fixed nanoparticles, where the direction of the magnetization vector in
the particle fluctuates, N eel’s superparamagnetism [11] is observed, characterized by
a change in the M€ ossbauer spectrum, provided that the frequency of the thermal
fluctuations is sufficiently high. The M€ ossbauer spectrum is a resonant c-absorption
spectrum that involves the emission and absorption of c-rays from the excited states
of a nucleus. (The M€ ossbauer effect is very complex and an exact description is
possible only by quantum mechanics; hence, the following rather “plausible”
explanation must be used with extreme care.) When an excited nucleus emits a
c-quantum with energy in the range from a few kilo electron volts to 100 keV, it must
recoil to conserve momentum because the c-photon has a nonzero mass. Therefore,
the energy of the emitted c-photon is reduced by the recoil energy in the range of a
few 10
À3 eV. Inversely, the same is occurring during the absorption of a c-quantum,
and this leads to a broadening of the energy distribution of the emission and
absorption line; this in turn reduces the probability for resonance absorption, as the
linewidth of the emission or absorption levels are significantly less than 10
À5 eV.
However, by placing the emitting and absorbing nuclei in a crystal, the crystal lattice
is used for recoil. This reduces the recoil energy loss to a value that emission and
absorption lines remain extremely narrow so that resonance absorption is possible.
This phenomenon is used to detect extremely small energy shifts by moving either
source or absorber with velocities on the order of a few millimeters per second. As an
186j 8 Magnetic Properties of Nanoparticles
material is reduced.
When considering superparamagnetism, distinction must be made between two
different definitions: (i) an older definition, which stems from Elmore [13] and is based
on the Brownian motion of magnetic particles in a liquid, and (ii) a younger definition,
as reported by N eel [11]. In the Brownian case, the particles are rotating in a liquid
when the direction of the magnetic field is changed. Based on these mechanisms, the
Brownian case is often referred to as “extrinsic” superparamagnetism, in contrast to
the “intrinsic” superparamagnetism in N eel’s case, which is the one discussed in the
preceding part of this chapter. The relaxation time (the time needed by the particle to
follow a change in the direction of the magnetic field) for the Brownian superparamagnetism is t B ¼ 3vg=kT, where g represents the viscosity of the liquid carrier
(the definitions of the other letters are outlined above). In the case of N eel’s superparamagnetism, the relaxation time is calculated using Eq. (8.11) [14]. The N eel case
will be discussed in detail later in this chapter. Both cases have in common that each
particle consists of a single domain and, provided that the measuring time constant is
adequate, hysteresis is not observed. Additionally, in both cases, the magnetization
curves follow Langevin’s law. When comparing the relaxation times for both cases, for
10-nm particles, in the Brownian case one finds values in the region of microseconds,
whereas in N eel’s case the relaxation time is about 1 ns. For magnetic particle with
sizes of approximately 1 mm, suspended in a liquid, the Brownian relaxation times are
close to 1 s. The relaxation time rules the ability of the particles to follow a change of an
external magnetic field with frequency f. In the case of t < 1/f, the particle is able to
follow the external frequency, and therefore remanence and coercivity are zero. In the
case of t > 1/f, hysteresis is observed.
In spatially fixed nanoparticles, where the direction of the magnetization vector in
the particle fluctuates, N eel’s superparamagnetism [11] is observed, characterized by
a change in the M€ ossbauer spectrum, provided that the frequency of the thermal
fluctuations is sufficiently high. The M€ ossbauer spectrum is a resonant c-absorption
spectrum that involves the emission and absorption of c-rays from the excited states
of a nucleus. (The M€ ossbauer effect is very complex and an exact description is
possible only by quantum mechanics; hence, the following rather “plausible”
explanation must be used with extreme care.) When an excited nucleus emits a
c-quantum with energy in the range from a few kilo electron volts to 100 keV, it must
recoil to conserve momentum because the c-photon has a nonzero mass. Therefore,
the energy of the emitted c-photon is reduced by the recoil energy in the range of a
few 10
À3 eV. Inversely, the same is occurring during the absorption of a c-quantum,
and this leads to a broadening of the energy distribution of the emission and
absorption line; this in turn reduces the probability for resonance absorption, as the
linewidth of the emission or absorption levels are significantly less than 10
À5 eV.
However, by placing the emitting and absorbing nuclei in a crystal, the crystal lattice
is used for recoil. This reduces the recoil energy loss to a value that emission and
absorption lines remain extremely narrow so that resonance absorption is possible.
This phenomenon is used to detect extremely small energy shifts by moving either
source or absorber with velocities on the order of a few millimeters per second. As an
186j 8 Magnetic Properties of Nanoparticles