The Periodic Transport
201
and when p = p 0 + dp, we have
k =
q
p
∂B y
∂x
=
q
p 0
∂B y
∂x
1
1 + dp/p 0
= 1 k 0 (1 − δ) ,
k 0 =
q
p 0
∂B y
∂x
, δ =
dp
p 0
.
For a small distance ds, we have
d ˆ
M =
1 0
−kds 1
.
And by defining
d ˆ
M 0 =
1
0
−k 0 ds 1
,
we have
ˆ
M = d ˆ
M d ˆ
M
−1
0
cos(μ 0 ) + α sin(μ 0 )
β sin(μ 0 )
−γ sin(μ 0 )
cos(μ ◦ ) − α sin(μ ◦ )
=
1
0
− (k − k 0 ) ds 1
cos(μ 0 ) + α sin(μ 0 )
β sin(μ 0 )
−γ sin(μ ◦ )
cos(μ ◦ ) − α sin(μ ◦ )
=
1 0
δk 0 ds 1
cos(μ 0 ) + α sin(μ 0 )
β sin(μ 0 )
−γ sin(μ 0 )
cos(μ 0 ) − α sin(μ 0 )
.
Using the normalization transformation
ˆ
A =
1/
√
β
0
α/
√
β
√
β
and ˆ
A
−1 =
√
β
0
−α/
√
β 1/
√
β
,
we obtain the transfer matrix in the normalized space as
ˆ
M = ˆ
A ˆ
M ˆ
A
−1
= ˆ
A
1 0
δk 0 ds 1
ˆ
A
−1
· ˆ
A
cos(μ 0 ) + α sin(μ 0 )
β sin(μ 0 )
−γ sin(μ 0 )
cos(μ 0 ) − α sin(μ 0 )
ˆ
A
−1
=
1
0
δβk 0 ds 1
cos(μ 0 ) sin(μ 0 )
− sin(μ 0 ) cos(μ 0 )
=
cos(μ 0 )
sin(μ 0 )
δβk 0 ds cos(μ 0 ) − sin(μ 0 ) cos(μ 0 ) + δβk 0 ds sin(μ 0 )
,
and
cos(μ) =
1
2
tr
ˆ
M
= cos(μ ◦ ) +
1
2
δβk 0 ds sin(μ 0 ),
μ = μ 0 + dμ =⇒ cos(μ) = cos(μ 0 ) − dμ sin(μ 0 ) =⇒ dμ = −
1
2
δβk 0 ds
=⇒ dν =
dμ
2π
= −
1
4π
δβk 0 ds =⇒ ξ N =
dν
dp/p 0
= −
1
4π
β(s)k 0 (s)ds.