316
F. Vanderveken et al.
rather flat dispersion relation interact with waves with a steep dispersion relation
near the crossing point.
12.4.2.3 Arbitrary Propagation Direction
We now consider an arbitrary propagation direction of the magnetoelastic wave with
respect to the equilibrium magnetization. In this case, the magnetoelastic body force
interacts with all displacement components. Conversely, all displacement components generate magnetic fields that interact with the magnetization. Hence, all magnetization and displacement components become coupled with each other. Again,
nontrivial wave-like solutions only exist when the determinant of the linear system
in (12.68) is zero, which leads to the dispersion relation
(ω
2
− ω
2
l )[(ω
2
− ω
2
t )
2
(ω
2
− ω
2
fm )
−(ω
2
− ω
2
t )J k
2
(ω fx cos
2
(θ ) + ω fy cos
2
(2θ)) − J
2 k
4 cos
2
(2θ) cos
2
(θ )]
−(ω
2
− ω
2
t )J k
2
[ω fy (ω
2
− ω
2
t ) sin
2
(2θ) + J k
2 sin
2
(2θ) cos
2
(θ )] = 0 .
(12.81)
Note that for θ = π/4, the coupling between the magnetic and the longitudinal elastic
wave reaches a maximum, whereas for θ = 0 and θ = π/2, this coupling is zero.
The dispersion relations of the resulting magnetoelastic waves are plotted in
Fig. 12.7 for material parameters of Ni and θ = π/6. For each frequency, multiple
magnetoelastic waves exist with different wavelengths. Since the system of equations is reduced to a set of linear differential equations by assuming weak dynamic
components, every linear combination of these different magnetoelastic waves is
also a solution of the system. The waves can be excited by dynamic magnetic fields
and/or mechanical forces. Therefore, it is possible to generate elastodynamics via
the magnetization or, vice versa, magnetization dynamics via the displacement in
magnetostrictive materials.
It can also be seen from the dispersion relations that the group velocity of the
magnetoelastic waves is different from the group velocity of the magnetic and elastic
waves. As mentioned earlier, the group velocity is defined as v g = ∂ω/∂k and thus
proportional to the slope of the dispersion relation. Hence, near the anticrossing, this
change in group velocity is most pronounced. On the other hand, the group velocity
of quasi-elastic and quasi-magnetic waves is nearly the same as their purely elastic
and magnetic counterparts, respectively.
The total energy of a magnetoelastic wave consists of several contributions. The
magnetic energy contribution is determined by the dynamic components m x and
m y . In this chapter, only dipolar, and exchange energy interactions were considered,
although other magnetic interactions, such as the magnetocrystalline [40, 41, 50] or
the Dzyaloshinskii–Moriya interaction [51] may also contribute to the total energy.
The magnetic energy is complemented by the energy of the elastic waves, which consists of both elastic and kinetic energy contributions and is fully determined by the
F. Vanderveken et al.
rather flat dispersion relation interact with waves with a steep dispersion relation
near the crossing point.
12.4.2.3 Arbitrary Propagation Direction
We now consider an arbitrary propagation direction of the magnetoelastic wave with
respect to the equilibrium magnetization. In this case, the magnetoelastic body force
interacts with all displacement components. Conversely, all displacement components generate magnetic fields that interact with the magnetization. Hence, all magnetization and displacement components become coupled with each other. Again,
nontrivial wave-like solutions only exist when the determinant of the linear system
in (12.68) is zero, which leads to the dispersion relation
(ω
2
− ω
2
l )[(ω
2
− ω
2
t )
2
(ω
2
− ω
2
fm )
−(ω
2
− ω
2
t )J k
2
(ω fx cos
2
(θ ) + ω fy cos
2
(2θ)) − J
2 k
4 cos
2
(2θ) cos
2
(θ )]
−(ω
2
− ω
2
t )J k
2
[ω fy (ω
2
− ω
2
t ) sin
2
(2θ) + J k
2 sin
2
(2θ) cos
2
(θ )] = 0 .
(12.81)
Note that for θ = π/4, the coupling between the magnetic and the longitudinal elastic
wave reaches a maximum, whereas for θ = 0 and θ = π/2, this coupling is zero.
The dispersion relations of the resulting magnetoelastic waves are plotted in
Fig. 12.7 for material parameters of Ni and θ = π/6. For each frequency, multiple
magnetoelastic waves exist with different wavelengths. Since the system of equations is reduced to a set of linear differential equations by assuming weak dynamic
components, every linear combination of these different magnetoelastic waves is
also a solution of the system. The waves can be excited by dynamic magnetic fields
and/or mechanical forces. Therefore, it is possible to generate elastodynamics via
the magnetization or, vice versa, magnetization dynamics via the displacement in
magnetostrictive materials.
It can also be seen from the dispersion relations that the group velocity of the
magnetoelastic waves is different from the group velocity of the magnetic and elastic
waves. As mentioned earlier, the group velocity is defined as v g = ∂ω/∂k and thus
proportional to the slope of the dispersion relation. Hence, near the anticrossing, this
change in group velocity is most pronounced. On the other hand, the group velocity
of quasi-elastic and quasi-magnetic waves is nearly the same as their purely elastic
and magnetic counterparts, respectively.
The total energy of a magnetoelastic wave consists of several contributions. The
magnetic energy contribution is determined by the dynamic components m x and
m y . In this chapter, only dipolar, and exchange energy interactions were considered,
although other magnetic interactions, such as the magnetocrystalline [40, 41, 50] or
the Dzyaloshinskii–Moriya interaction [51] may also contribute to the total energy.
The magnetic energy is complemented by the energy of the elastic waves, which consists of both elastic and kinetic energy contributions and is fully determined by the