Chemistry Reference
In-Depth Information
distance determination at this particular temperature. The distances obtained by using 1/
T
M
-1/
T
M0
for canthaxanthin at 125 K and for BI at 40 K are found to be 13.0 and 10.5 Å, respectively.
To measure distances in the wider temperature range, this procedure was modii ed. Relaxation
of the carotenoid occurs through several different mechanisms including the dipolar-dipolar inter-
action. Assuming that
k
dd
is the rate constant of the dipolar-dipolar interaction and
K
=
(
k
1
+
k
2
+
k
3
…) is the sum of the rate constants of all other relaxation pathways, we can extract
k
dd
from the
following equation:
+
−
(
k
)
+
K
t
e
dd
(9.27)
−
kt
e
=
dd
−
Kt
e
W
(
t
) was calculated from Equation 9.28 by numerical integration over the angle between the exter-
nal magnetic i eld and the inter-nuclear axis (
θ
), at any given instant
τ
:
π
2
⎧
2
⎫
2
⎛
τ
⎞
⎪
⎡
sinh(
R
τ
)
⎤
D
⎪
∫
2
(9.28)
W
( )
τ=
exp
−
∗
d
θ
sin
θ
cosh(
R
τ+
)
+
sinh (
R
τ
)
⎨
⎬
⎜
⎟
⎢
⎥
2
T
2
RT
4
R
⎝
⎠
⎣
⎦
⎪
⎪
1
⎩
1
⎭
0
In Equation 9.28,
D
and
R
were calculated from Equations 9.29 and 9.30:
h
μβ
gg
2
012
2
(9.29)
D
=
(1
−
3 cos
θ
)
3
4
π
(9.30)
2
−
=−
2
2
4
RT D
1
where
T
1
is the longitudinal relaxation time of the fast relaxing Ti
3+
ion
D
is the dipole-dipole interaction between the slow relaxing carotenoid radical and the fast
relaxing Ti
3+
ion
r
is the interspin distance
θ
is the angle between the direction of the external magnetic i eld and a vector connecting the
two species with
g
-values
g
1
and
g
2
Prior to the integration, a change of variable was carried out by setting
x
=
cos
θ
, where
θ
∈
[0,
π
/2]
and
x
∈
[0, 1], and Equation 9.28 was transformed into Equation 9.31:
1
⎧
2
⎫
2
⎛
τ
⎞
⎪
⎡
sinh(
R
τ
)
⎤
D
⎪
∫
2
(9.31)
W
( )
τ=
exp
−
∗
dx
cosh(
R
τ+
)
+
sinh (
R
τ
)
⎨
⎬
⎜
⎟
⎢
⎥
2
T
2
RT
4
R
⎝
⎠
⎣
⎦
⎪
⎪
1
⎩
1
⎭
0
In Equation 9.31,
D
is related to
x
by
2
h
μβ
gg
012
2
(9.32)
D
=
(1
−
3
x
)
3
4
π
The integration was carried out with the extended trapezoidal rule for an integral over function
f
(
x
)
b
⎛
fx
()
fx
()
⎞
⎛
b a
−
⎞
∫
0
n
I
=
f x dx
()
≈
+
f x
( )
+
+
f x
(
)
+
∗
(9.33)
⎜
⎟
⎜
⎟
1
n
−
1
2
2
n
⎝
⎠
⎝
⎠
a
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