Chemistry Reference
In-Depth Information
−
S,...,S,
and the energy levels of the magnetic molecule are given by
Dm
2
Bm
4
ε
m
=−
−
−
gμ
B
H
z
m
(141)
The transition frequency for a pair of levels is
m
m
m
D
m
2
m
2
B
gμ
B
H
z
(142)
ε
m
=−
m
ω
mm
=
ε
m
−
−
+
+
+
+
m
(levels on different sides of the barrier cre-
ated by the uniaxial anisotropy) defines the resonance values of the longitudinal
field
H
z
.
For the generic model of molecular magnets with
B
Condition
ω
mm
=
0 for
m/
=
=
0 the latter are
given by
gμ
B
H
z
=
kD,
k
=
0
,
±
1
,
±
2
,...
(143)
For these fields,
all
levels in the right well
m
=−
m
−
k
(144)
are at resonance with the corresponding levels in the left well
m<
0
.
For the
realistic model with
B>
0
,
the field creating resonances between low-lying levels
with large
m
2
m
2
is greater than the resonance fields for high levels. Transverse
anisotropy and transverse field
H
x
that enter
H
A
result in the tunneling under the
barrier and tunneling splitting of the resonant levels
m, m
.
Note that the resonance
condition, Eq. (143) does not depend on the transverse field. To the contrary, for
B/
+
=
0 the resonance condition depends on the transverse field.
For Mn
12
used in illustrations below, we adopt the values
D/k
B
=
0
.
548 K and
10
−
3
K [14-16] that make up the barrier of 66 K. Here
H
A
in Eq.
(140) can contain second- and fourth-order transverse anisotropy,
B/k
B
=
1
.
1
×
E
S
x
−
S
y
H
A
=
C
(
S
4
S
4
−
+
+
+
)
(145)
10
−
5
For Mn
12
,
C/k
B
=
0 in the ideal case be-
cause of the tetragonal symmetry of the crystal. However, it was shown [16] that
local
molecular environments of Mn
12
molecules have a twofold symmetry and
rotated by 90
◦
for different molecules. Although on average the fourfold sym-
metry of the crystal is preserved, it gives rise to nonzero
E
that will be set to
E/k
B
=
3
×
K [14-16], whereas
E
=
10
−
3
K.
The spin Hamiltonian of molecular magnets can be easily numerically diago-
nalized to yield eigenstates
2
.
5
×
|
α
and transition frequences
ω
αβ
,α,β
=
1
, ..,
2
S
+
1
.
