CHESS & LEPP 103 [email protected]Introduction to Accelerator Physics Fall semester 2017 ρ κ κ κ 1 with = − = e e s ds d ! ! ds R d = ! κ κ e e e e e e s R e s b s ds d s ds d ds d s ! ! ! ! ! ! ! ! × ≡ − ≡ ≡ ) ( b s ds d e T e e ! ! ! ' + = κ κ ( ) κ κ κ − ⋅ = ⋅ = e e e e ds d s s ds d ! ! ! ! 0 κ e T e b ds d ! ! ' − = ( ) ' 0 T e e e e b ds d b ds d + ⋅ = ⋅ = ! ! ! ! κ κ ) ( s R ! ) ( ) ( ) ( s e y s e x s R r y x ! ! ! ! + + = s b e x e T x y e T y x r ! ! ! ! ) 1 ( ) ' ' ( ) ' ' ( ' κ κ + + + + − = Accumulated torsion angle T The Frenet Coordinate System
9
Embed
R s κ with κ ! ! ! ! 0 e −κhoff/LECTURES/17F...CHESS & LEPP 103 [email protected] Introduction to Accelerator Physics Fall semester 2017 ρ κ κ κ 1 ds d e s =
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
CHESS & LEPP
103
[email protected] Introduction to Accelerator Physics Fall semester 2017
ρκκ κ
1with =−= eesdsd !!
dsRd =!
κ
κ
eeeeee
sRe
sb
sdsd
sdsd
dsd
s
!!!
!!!
!!
×≡
−≡
≡ )(
bsdsd eTee !!! '+=κκ
( ) κκκ −⋅=⋅= eeee dsd
ssdsd !!!!0
κeTebdsd !! '−=
( ) '0 Teeee bdsd
bdsd +⋅=⋅=
!!!!κκ
)(sR!
)()()( seysexsRr yx!!!!
++=
sb exeTxyeTyxr !!!! )1()''()''(' κκ ++++−=
Accumulated torsion angle T
The Frenet Coordinate System
CHESS & LEPP
104
[email protected] Introduction to Accelerator Physics Fall semester 2017
yyxxsdsd eee !!!
κκ −−=
)cos()sin()sin()cos(TeTeeTeTee
by
bx!!!
!!!
+≡
−≡
κ
κ
sysydsd
sxsxdsd
eeTeeeTe!!!
!!!
κκ
κκ
==
==
)sin()cos(
)(sR!
)()()( seysexsRr yx!!!!
++=
dds!r = x ' !ex + y '
!ey + (1+ xκ x + yκ y )!es
The Curvi-linear System
CHESS & LEPP
105
[email protected] Introduction to Accelerator Physics Fall semester 2017
ppsrsrsph
mdtd
dsd !!"!"!
=== −−γ111
FFsps
epppeppeppp
sphm
dtd
syyxxsyysyxxsxdsd
!!"!"
!!!!
γ
κκκκ
===
+++−+−=−− 11
''' )()()(
s
h
yxyxdsd eyxeyexr !
""#""$%!!! )1('' κκ ++++=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+
+=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
ysyphm
xsxphm
yph
xph
y
x
pFpFpp
ppyx
s
s
s
s
κ
κγ
γ
'
'
''
sphmst γ== −1' !
FppEpcpmcpcE
sph
dsd
dsd
dpd
!!!!
⋅==+= 2222 )()('
222 )()( mcpcE +=
Phase Space ODE
ddt!p =!F
CHESS & LEPP
106
[email protected] Introduction to Accelerator Physics Fall semester 2017
),,,,,( δτbyaxz =!
0
00
0
2
00
0
00
)()(,,,pEtt
vctt
EEE
pp
bppa yx −=−=
−=== τδ
01
00
00
00'
0
2
0
0
2
''
''
','',''
pKKKE
pKEKKt
pKbpKypKapKx
KpKx
vc
Et
vc
E
yb
xa
xx
px
ττ
δδ
δ
τ
−∂=∂−=⇒∂−=
∂=∂=⇒∂=−
⎩⎨⎧
−∂=∂=
−∂=∂=⇒
⎭⎬⎫
−∂=
∂=
−
New Hamiltonian:
0~ pKH =
Using a reference momentum p0 and a reference time t0:
Usually p0 is the design momentum of the beam And t0 is the time at which the bunch center is at “s”.
x'xa ≈
sΔ≈τ
6 Dimensional Phase Space
CHESS & LEPP
107
[email protected] Introduction to Accelerator Physics Fall semester 2017