R s κ with κ ! ! ! ! 0 e −κhoff/LECTURES/17F...CHESS & LEPP 103 [email protected]...
9
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
Transcript of R s κ with κ ! ! ! ! 0 e −κhoff/LECTURES/17F...CHESS & LEPP 103 [email protected]...
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
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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
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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
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),,,,,( 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
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Only magnetic fields: 0=E!
Mid-plane symmetry: ),,(),,(,),,(),,( syxBsyxBsyxBsyxB yyxx −=−−=
0',)('
')()1())(1(
'
''
11)(
21
1)'(
1)(
0
0
00
00
00000
02220
=−=−==
=⇒++−=++++−=
+−=+==
=⇒====
−
++
−−+
δκτ
κδκκδκκ
κκκκ
x
ykbkxxkx
Bqvpa
byax
pE
vh
vpE
dsttd
pp
ysvpx
pp
xdtd
vph
pepp
pp
ppdpp
ppp
s
s
s
s
s
xxs
x
yx
x
z
x
!!
δκκ xxyxkbaH −+−++= 222122
212
212
21 )(
Hamiltonian:
Simplified Equation of Motion
xksyxByksyxB qp
qp
yqp
x000 1),,(,),,( +== ρ
Linearization in :
0,0
0
0
0
0
0 →→ −−−vvv
EEE
pppHighly relativistic :
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zsFz !! )('=
0z!
Linear equation of motion:
Matrix solution of the starting condition 0)0( zz !!=
01drift )( zLMz !!=
01drift2quad )()( zLMLMz !!=
01drift2quad3drift )()()( zLMLMLMz !!=
01drift2quad3drift4bend )()()()( zLMLMLMLMz !!=
Bend
Quadrupole
Matrix Solutions
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⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
000
0
''''''
b
a
byax
δ
τ
0
0
0
0
00
00
1001
00
010
10
0010
1
zs
s
bsbyasax
byax
!
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
+
+
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
δ
τ
δ
τ
Note that in nonlinear expansion so that the drift does not have a linear transport map even though is completely linear.
ax ≠'
sxxsx '00)( +=
The Drift
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x ''+ x κ 2 = δκ
y '' = 0τ ' + xκ = 0
)sin()cos('' 2 sBsAxxx HHH κκκ +=⇒−=Homogeneous solution:
κδκκκκκκ
κκκκκκ
κκ
κδ
+−=+−−=
+++−=
+=
=
≡
xsBsAxx
sBsAsBsAxssBssAx
220
)cos(')sin('''
)sin(')cos(')cos()sin(')sin()()cos()(
!!!!! "!!!!! #$
!!!! "!!!! #$
Variation of constants:
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
− δκκ
κκ 0''
)cos()sin()sin()cos(BA
ssss
(natural ring focusing)
The Dipole Equation of Motion
CHESS & LEPP
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⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
−−−
−
−
=
−−
−
10])[sin(1
000
]1)[cos()sin(
010
10
)sin(0)]cos(1[0
0)cos()sin()sin()cos(
11
11
κκκκκκ
κ
κκκκκ
κκ κ
ssss
sss
ssss
M
)sin()cos(with)sin()cos(1 sBsAx
BA
ss
BA
H
H κκκ
κκδ +=⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟⎠
⎞⎜⎜⎝
⎛ −
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛
δκκ
κκ 0)cos()sin()sin()cos(
''
ssss
BA
κτ x−='
The Dipole
