Environmental Engineering Reference
In-Depth Information
interaction may originate from, for instance, the term
O
2
+
O
2
}
2
(
O
2
+
O
2
)
−
B
γ
2
{
−
cos 2
φ
(
γ
1
−
γ
1
)
in (5.4.6), or the corresponding terms in (5.4.16). In contrast to the lin-
ear couplings considered above, the symmetry-preserving
α
-strain part
of the magnetoelastic Hamiltonian makes a contribution to the quadratic
interaction terms. Using the procedure of Evenson and Liu (1969), it
is straightforward, if somewhat tedious, to relate the interaction am-
plitudes in eqn (5.4.50) to the magnetoelastic coupling parameters. We
shall not perform this analysis here, but refer instead to the detailed cal-
culations of Jensen (1971a,b). The interactions in eqn (5.4.50) have the
consequence that the equations of motion of the magnon Green func-
tion
α
q
;
α
q
involve new, higher-order mixed Green functions like
. Performing an RPA or Hartree-Fock decoupling, as in
(5.2.29), of the three-operator products which occur in the equations of
motion of the new Green functions, we obtain a closed expression for
the magnon Green function, which may be written
α
q
−
k
β
k
;
α
q
1
hω − E
q
(
T
)
−
Σ(
q
,ω
)
,
α
q
;
α
q
(5
.
4
.
51)
=
where Σ(
q
,ω
)isthe
self-energy
, due to the interactions in (5.4.50), of
the magnons of wave-vector
q
. Neglecting
V
ν
(
q
,
k
), we find that the
self-energy at
T
=0is
2
hω
+
ih − E
q
+
k
(0)
|
U
ν
(
k
,
q
)
|
Σ(
q
,ω
) = lim
→
0
+
.
(5
.
4
.
52)
−
hω
ν
k
k
ν
These interactions are not diagonal in reciprocal space and the magnons
are therefore affected by all the phonons.
Whenever
k
has a value
such that
E
q
(0)
E
q
+
k
(0) +
hω
ν
k
, the real part of the denomina-
tor in (5.4.52) vanishes close to the magnon pole at
q
, as determined
by (5.4.51). This implies a negative imaginary contribution to Σ(
q
,ω
),
when
hω
E
q
(0), and hence a reduction in the lifetime of the magnons.
The energy of the magnons at
q
is approximately given by
E
q
(0) +
Re
Σ(
q
,ω
)
,with
hω
E
q
(0). At non-zero temperatures, the self-
energy terms increase in proportion to the Bose population-factors of
the magnons and phonons involved. These interactions, quadratic in
the magnon operators, do not lead to the kind of hybridization effects
produced by the linear couplings, but rather give rise to a (small) renor-
malization of the normal-mode energies and to a finite lifetime of the ex-
citations. These effects are entirely similar to those due to the magnon-
magnon interactions appearing in the spin-wave theory in the third order
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