As we have shown in the previous chapter, the numerical evaluation of the energy barrier should be done in a multidimensional space and it is a difficult task. With this aim we use the Lagrange-multiplier method described in section 2.4. We evaluate the energy barriers of multi-spin particles (MSPs) by numerically
computing the difference between the energy at the saddle point
and at the minimum, and compare them with analytical results obtained from the EOSP approximation.
Note that in the wide range of the parameters several energy
barriers (corresponding to different paths of magnetization
rotation) coexist in the system in accordance with the complex
character of the effective potential with two competitive
anisotropies, see Figs. 2.3(c) and
2.9(c). In what follows, in most cases we
will only discuss the energy barrier which corresponds to the
change in direction. In the case corresponding to elongated nanoparticles cut from fcc lattice and uniaxial anisotropy in the core, see
Fig.2.9(d), the only energy barrier
corresponds to the rotation around direction.
Energy barrier as a function of for a spherical
particle cut from an sc lattice. The particle contains
spins and has the uniaxial core anisotropy
The solid line is a plot of the analytical expression
Fig. 3.2 shows the energy barrier of a spherical
particle cut from an sc lattice as a function of . The
nonlinear behavior of the energy barrier with follows
quantitatively that of the EOSP potential (3.7).
Indeed, the solid line in this plot is the analytical result
(3.8), using analytical expressions of Eqs. (2.14-2.15) together
with the pure core anisotropy contribution.
The discrepancy at the relatively large is due to the fact that the analytical expressions are valid only if the condition (2.22) is fulfilled; the core-surface mixing (CSM) contribution has not been taken into account.
Energy barriers against for truncated octahedral multi-spin particle cut from an fcc lattice (
) and spherical particles cut from the fcc (
) and hcp (
) lattices. Uniaxial core anisotropy with
In Fig. 3.3 we represent the energy barriers as a function of
for MSPs with different shapes and internal structures. First of
all, one can see a different dependence on as compared to
MSPs with the sc lattice. In the present case, i.e.,
energy barriers are given by
Eqs. (3.9)(a) and (c).
For small values of , for which
the energy barrier, in the first approximation neglecting the CSM
term, is independent of . Accordingly, the nearly constant
value of the energy barrier, coinciding with that of the core, is
observed for multi-spin particles in a large range of .
For larger , the energy barrier increases, since
multi-spin particles cut from an fcc lattice. At very large values of , i.e.,
the energy barriers depend approximately linearly on and may have
values larger than that inferred from the pure core anisotropy.
The energy barriers for ellipsoidal multi-spin particles are shown in Fig. 3.4.
Note that in this case the effective uniaxial anisotropy constant
is a linear function of , according to the analytical results (2.16), (2.17) and the numerical results presented in
In this case the value of the energy barrier is not symmetric with respect to the change of the sign of . This is due to the fact that for the effective uniaxial constant is a sum of the core anisotropy and the first-order contribution owing to elongation. On the contrary, when , the "effective core anisotropy"
is smaller than the pure core anisotropy
in Eq. (2.6).
This means that at some the effective uniaxial anisotropy constant (
) may change sign. At the same time the effective cubic anisotropy
remains positive and is proportional to . Accordingly, at the vicinity of the point at which
, rapid changes of the character of the energy landscape occur as we have shown in Fig. 2.10 in the previous chapter.
Energy barriers versus of ellipsoidal multi-spin particles with different aspect ratio (
, with uniaxial core anisotropy
. The solid lines are linear fits.
The analysis, based on the effective one spin problem potential shows that when
the energy barriers of ellipsoidal multi-spin particles are defined by Eqs. 3.9(a) and (c), and for negative
these are given by Eq. 3.9(b) and (c).
We would like to note that a regime of linear behavior in exists for both
and (see Fig. 3.4). In some region
of the effective anisotropy constants, e.g.,
, , the energy barrier
, i.e., it is
independent of the cubic contribution (neglecting again the CSM
term). Consequently, it is linear in , according to
Eq. (2.17). The interval of these parameters is especially
large in MSPs with for which
does not change sign.