Chemistry Reference
In-Depth Information
The primary integral gives
ς
dθ
dz
2
+sin
2
θ
=
k
−
2
,
(6.8)
where
k
2
<
1 and
k
−
2
is the integral constant. The further integral gives
the relation of
θ
and
z
as
=
θ
0
dθ
1
k
z
ς
=
F
(
θ, k
)
,
(6.9)
k
2
sin
2
θ
)
2
(1
−
where
F
(
θ, k
) is the elliptic integral of the first kind. In the absence of
a magnetic field, no deformation occurs and the twist angle is a linear
function of
z
. But as a magnetic field is applied, the dependence of
θ
on
z
is no longer a linear function. The Elliptic integral of the first kind
F
(
θ, k
)
is a periodic function with the period 4
K
(
k
) where
K
(
k
)
F
(
π/
2
,k
), the
complete elliptic integral of the first kind. Thus the helical pitch
P
for a
magnetic field
H
is
≡
P
=4
kςK
(
k
)
(6.10)
and the integral constant
k
is expressed implicitly by
E
(
k
)
k
=
π
2
ςP
0
,
(6.11)
where
E
(
k
)=
π/
2
0
(1
−
k
2
sin
2
θ
)
2
dθ
, is the complete elliptic integral of the
second kind. Substituting it into Equation 6.10 then yields the ratio of the
current to intrinsic helical pitch as
P/P
0
=
2
π
2
K
(
k
)
E
(
k
)
(6.12)
If
k
is small, the ratio is rewritten approximately as
P
0
ς
4
P
0
≈
1+
1
P
.
(6.13)
32
It is noted that for a small magnetic field, the helical pitch increases
slowly until
k
2
= 1 at which one has
E
(
k
)=1and
K
(
k
)
→∞
and then
P/P
0
→∞
.
(6.14)
