Chemistry Reference
In-Depth Information
Spin polarization in our low-temperature tunneling process is given by
S
S
z
=−
Sρ
+
mρ
mm
(265)
−
S,
−
S
m
m
=
1 decay faster than
m
,
their populations
are small and thus their contribution into the above formula can be neglected. Then
one can write
m
+
As the states with
m
=
1
,...,S
−
m
ρ
m
m
+
S
z
=−
Sρ
+
Sρ
SS
−
S,
−
S
m
ρ
m
m
+
=−
Sρ
+
S
(1
−
ρ
−
ρ
m
m
)
−
S,
−
S
−
S,
−
S
m
)
ρ
m
m
=
S
−
2
Sρ
−
(
S
−
(266)
−
S,
−
S
As both
ρ
S
and
ρ
m
m
go asymptotically to zero, the magnetization change in
−
S,
−
S
z
0
−
S
z
∞
=−
the process is
2
S
, and
m
)
ρ
m
m
S
z
t
−
S
z
∞
=−
2
Sρ
−
(
S
−
(267)
−
S,
−
S
Thus the integral relaxation time is given by
0
dt
dt
ρ
ρ
m
m
(
t
)
(268)
∞
m
2
S
S
z
t
−
S
z
∞
S
−
τ
int
=
=
S
(
t
)
+
−
S,
−
S
z
0
−
S
z
∞
0
For a small bias, say,
m
=
1 and large spin
,
the contribution of the second
term is small. Using Eq. (263), one obtains
∞
S
−
2
2
i
2
W
i
2
W
1
−
λ
−
λ
−
+
dtρ
S
(
t
)
=
+
−
S,
−
2
2Re(
λ
)
2Re(
λ
)
|
λ
+
−
λ
−
|
0
+
−
2Re
2
W
−
λ
+
i
2
W
−
λ
−
−
−
(269)
λ
−
λ
+
+
and
4
2
∞
2
2
i
2
W
i
2
W
−
λ
−
λ
−
+
dtρ
m
m
(
t
)
=
2
|
λ
+
−
λ
−
|
0
1
2Re(
λ
1
2Re(
λ
2
×
)
+
)
−
Re
(270)
λ
−
λ
+
+
+
−
Mathematica-aided simplification yields
∞
2
m
+
1
,m
m
+
1
,m
2
2
4
W
+
+
dtρ
S
(
t
)
=
(271)
−
S,
−
0
