Chemistry Reference
In-Depth Information
the phonon frequency. The operator
δ
φ
that follows from Eq. (148) is given by
a
k
λ
+
k
λ
2
MN
k
λ
e
k
λ
]
e
i
k
·
r
√
ω
k
λ
1
2
[
i
k
×
a
†
−
δ
φ
=
(153)
k
,
survive in this formula. Whereas
V
(1)
is linear
in phonon operators and describes direct phonon processes,
V
(2)
is quadratic, and
describes Raman processes. Relaxation rates due to Raman processes are generally
much smaller than that due to the direct processes since they are the next order in
the spin-phonon interaction. However, the rates of direct processes can be small
for special reasons, then Raman processes become important. Processes of orders
higher than Raman always can be neglected.
It is important that the spin-phonon interaction above does not include any
poorly known spin-lattice coupling coefficients and it is entirely represented by
the crystal field
H
A
.
Moreover, the relaxation terms in the DME can be represented
in the form that does not explicitly contain
H
A
,
the information about it being ab-
sorbed in the spin eigenstates
Only transverse phonons,
e
k
λ
⊥
and transition frequencies
ω
αβ
that can be found
by numerical diagonalization of
H
S
.
This can be achieved either by changing from
the laboratory frame to the local lattice frame in which
H
A
remains constant, but
an effective rotation-generated magnetic field arises [12, 13, 17] or by manipu-
lating matrix elements of the spin-phonon interaction with respect to spin states,
|
α
α
V
β
, [13]. Both methods are mathematically equivalent [13]. In particular,
for
V
(1)
one can use
H
A
,
S
=
H
S
−
H
Z
,
S
=
H
S
,
S
+
i
S
×
gμ
B
H
(154)
are eigenstates of
H
S
to obtain the spin matrix element
|
and the fact that
α
i
α
H
A
,
S
β
=
(1)
αβ
≡
i
ω
αβ
α
|
S
|
β
−
α
|
S
|
β
×
gμ
B
H
(155)
For
V
(2)
, one writes [18]
H
A
,S
ξ
,S
ξ
=
H
S
−
H
Z
,S
ξ
,S
ξ
⎛
⎞
δ
ξξ
ξ
=
H
S
,S
ξ
,S
ξ
−
⎝
H
ξ
S
ξ
−
⎠
(156)
gμ
B
H
ξ
S
ξ
