144
H. Khurshid et al.
hollow MNPs. As the size of the nanoparticles decreases, the slope of the M–H
curves tends to increase indicating that the proportion of disordered surface spins
increases with decreasing size of the hollow MNPs. As the temperature increases,
the M–H loops resemble more and more those of magnetic nanoparticles with superparamagnetic + paramagnetic behavior. The superparamagnetic behavior is related
to the core spins of the nanograins in the shell, while the paramagnetic behavior can
be related to the surface spins of these nanograins. Their relative contribution can
vary depending upon the size and number of nanograins (size and shell thickness).
6.3.2 Surface Anisotropy and Spin Disorder
Since both inner and outer surfaces of a hollow nanoparticle contribute to enhance
its total surface area and hence surface anisotropy K s , this contributes to increase
the effective anisotropy (K eff ) of the whole system via: K eff = K c + 6K s /D, where
K c is the anisotropy associated with the core spins and D is the mean diameter of
the nanoparticle. The effective anisotropy of these MNPs can be estimated from
the blocking temperature, using the standard formula K eff V = 25k B T B , where T B
corresponds with blocking temperature and V with the volume of the nanograins in
the shell. For hollow γ-Fe 2 O 3 MNPs, effective anisotropies around ~10
6–7 erg/cm
3
have been reported [13, 15, 20, 52, 53], which can be up to two orders of magnitude
higher than those corresponding to solid maghemite nanoparticles (~10
5 erg/cm
3 ).
Other estimations of the effective anisotropy, for example by means of AC susceptibility measurements, tend to yield similar values. This increase of the effective
anisotropy can be attributed to the enhanced contribution from both surface and
finite-size effects in hollow MNPs. The spins lying at the surface and interface of the
magnetic nanograins can give rise to a great enhancement of the surface anisotropy.
To quantify the surface spins contribution toward magnetic properties, we can
analyze the magnetization vs magnetic field, M–H, loops. In an ensemble of MNPs,
the uncompensated surface spins are well known to provide a linear contribution
(paramagnetic) to the magnetization. To extract paramagnetic contribution to the
magnetization, the experimental M–H data can be fitted to the Langevin function
with an added linear term (6.1)
M(H ) = M
MSP
S
[coth
μH
K T
−
K T
μH
] + χ
PM H
(6.1)
where M S
SPM is the saturation magnetization of the SPM part (corresponding to the
“core” spins inside the shell), μ is the average magnetic moment of SPM particles,
and χ
PM is the susceptibility of the paramagnetic contribution (corresponding to the
“surface” spins at the shell) that is linear with the magnetic field H.
As an example, in Fig. 6.5 we present the M–H loops and corresponding fittings
to (6.1) for 9 and 18 nm hollow γ-Fe 2 O 3 MNPs. As depicted, for 9 nm hollow MNP
H. Khurshid et al.
hollow MNPs. As the size of the nanoparticles decreases, the slope of the M–H
curves tends to increase indicating that the proportion of disordered surface spins
increases with decreasing size of the hollow MNPs. As the temperature increases,
the M–H loops resemble more and more those of magnetic nanoparticles with superparamagnetic + paramagnetic behavior. The superparamagnetic behavior is related
to the core spins of the nanograins in the shell, while the paramagnetic behavior can
be related to the surface spins of these nanograins. Their relative contribution can
vary depending upon the size and number of nanograins (size and shell thickness).
6.3.2 Surface Anisotropy and Spin Disorder
Since both inner and outer surfaces of a hollow nanoparticle contribute to enhance
its total surface area and hence surface anisotropy K s , this contributes to increase
the effective anisotropy (K eff ) of the whole system via: K eff = K c + 6K s /D, where
K c is the anisotropy associated with the core spins and D is the mean diameter of
the nanoparticle. The effective anisotropy of these MNPs can be estimated from
the blocking temperature, using the standard formula K eff V = 25k B T B , where T B
corresponds with blocking temperature and V with the volume of the nanograins in
the shell. For hollow γ-Fe 2 O 3 MNPs, effective anisotropies around ~10
6–7 erg/cm
3
have been reported [13, 15, 20, 52, 53], which can be up to two orders of magnitude
higher than those corresponding to solid maghemite nanoparticles (~10
5 erg/cm
3 ).
Other estimations of the effective anisotropy, for example by means of AC susceptibility measurements, tend to yield similar values. This increase of the effective
anisotropy can be attributed to the enhanced contribution from both surface and
finite-size effects in hollow MNPs. The spins lying at the surface and interface of the
magnetic nanograins can give rise to a great enhancement of the surface anisotropy.
To quantify the surface spins contribution toward magnetic properties, we can
analyze the magnetization vs magnetic field, M–H, loops. In an ensemble of MNPs,
the uncompensated surface spins are well known to provide a linear contribution
(paramagnetic) to the magnetization. To extract paramagnetic contribution to the
magnetization, the experimental M–H data can be fitted to the Langevin function
with an added linear term (6.1)
M(H ) = M
MSP
S
[coth
μH
K T
−
K T
μH
] + χ
PM H
(6.1)
where M S
SPM is the saturation magnetization of the SPM part (corresponding to the
“core” spins inside the shell), μ is the average magnetic moment of SPM particles,
and χ
PM is the susceptibility of the paramagnetic contribution (corresponding to the
“surface” spins at the shell) that is linear with the magnetic field H.
As an example, in Fig. 6.5 we present the M–H loops and corresponding fittings
to (6.1) for 9 and 18 nm hollow γ-Fe 2 O 3 MNPs. As depicted, for 9 nm hollow MNP