Chemistry Reference
In-Depth Information
In the secular approximation, one needs the rate
α
=
α
=
2
(2)
D
2
(2)
αα
R
(2)
αα,α
α
2
(2)
Q
(2)
(2)
αα
,ξξ
(2)
α
α,ξξ
¯
¯
=
αα,α
α
=
ξξ
2
(2)
D
2
2
(2)
D
2
2
(2)
(2)
∗
(2)
αα
,ξξ
¯
¯
¯
=
αα
,ξξ
=
(193)
αα
,ξξ
ξξ
ξξ
where Eq. (194) follows from Eqs. (158) and (186):
2
γ
ε
α
2
ε
γ
α
S
ξ
γ
γ
S
ξ
β
α
S
ξ
γ
γ
S
ξ
β
1
(2)
αβ,ξξ
¯
≡−
+
ε
β
−
+
2
gμ
B
H
ξ
α
S
ξ
β
+
H
ξ
α
S
ξ
β
1
+
δ
ξξ
ξ
gμ
B
H
ξ
α
S
ξ
β
−
(194)
Integration in Eq. (191) is limited by the Brillouin zone, so that
ω
k
does not
exceed some maximal value. We will use the Debye model in which the phonon
spectrum continues in the same form up to the Debye frequency
D
which is the
upper bound of integration. Thus Eq. (191) can be represented in the form
G
6
D
T
D
2
24
2
π
3
(
k
B
T
)
7
E
t
(2)
=
(195)
where
y
x
n
e
x
(
e
x
G
n
(
y
)
≡
dx
(196)
1)
2
−
0
For
T
D
, the integration can be extended to infinity. Using
G
6
(
∞
)
=
16
π
6
/
21
,
one obtains
π
3
D
2
(
k
B
T
)
7
756
(2)
=
(197)
E
t
D
one can use
G
6
(
y
)
=
y
5
/
5 that yields
On the contrary, for
T
D
2
2880
π
3
(
k
B
D
)
5
(
k
B
T
)
2
E
t
(2)
=
(198)
The transition between these two regimes takes place at
T/
D
,
1
=
1
/y
≈
21
/
5
16
π
6
1
/
5
0
.
2
,
that is, much lower than the Debye temperature. For
this reason, the contribution of Raman processes is small in comparison to that
×
≈
