Lecture 8
ACCELERATOR PHYSICS
Melbourne
E. J. N. Wilson
Lecture 8 - E. Wilson - 3/20/2008 - Slide 1
Lecture8 – Accelerating Cavities Icontents
‹ Necessary
conditions for acceleration
‹ Waves in free space
‹ Two travelling waves in a guide.
‹ A transverse electric (H) mode
‹ Phase velocity and Group velocity
‹ Transverse magnetic modes
‹ Transit time factor
‹ The cylindrical cavity
Lecture 8 - E. Wilson - 3/20/2008 - Slide 2
Recap of previous lecture
Longitudinal dynamics II
‹ Transition
- does an accelerated particle
catch up - it has further to go
‹ Phase jump at transition
‹ Synchrotron motion
‹ Synchrotron motion (continued)
‹ Large amplitudes
‹ Buckets
‹ Adiabatic capture
‹ A chain of buckets
Lecture 8 - E. Wilson - 3/20/2008 - Slide 3
Maxwell forbids this!
dB
∇×E= −
dt
‹Become in its integral form
∂
∫ΓE.ds = − ∂t ∫SB.nda
‹Hence
there can be no acceleration
without time dependent magnetic field
‹We also see how time dependent flux
may accelerate
‹
BattAcc.gif
Lecture 8 - E. Wilson - 3/20/2008 - Slide 4
Waves in free space
v =c =
v=
1
ε 0 μ0
1
ε 0 Ke μ 0 Km
Km
E
= 376.6
ohms
Ke
H
P = (E × H) watts.m
Lecture 8 - E. Wilson - 3/20/2008 - Slide 5
-2
Two travelling waves in a guide.
Lecture 8 - E. Wilson - 3/20/2008 - Slide 6
A transverse electric (H) mode
(formed by superposition of the two waves)
1
λg
2
Lecture 8 - E. Wilson - 3/20/2008 - Slide 7
=
1
λ
2
−
1
λc 2
Group velocity
E = E0 sin[(k + dk ) x − (ω + dω )t]
+E0 sin[(k − dk ) x − (ω − dω )t]
= 2E0 cos[kx − ωt]sin[dkx − dωt]
f1 (x,t) = sin[kx − ωt]
kx − ω t = const
df ∂f
∂f
= +v
dt ∂t
∂x
k = 2π / λ
∂f1 (x,t) / ∂t ω
vp = −
=
∂f1 (x,t ) / ∂x k
f2 (x,t) = sin[dkx − dω t]
xδk − tδω
Lecture 8 - E. Wilson - 3/20/2008 - Slide 8
∂f2 (x,t) / ∂t dω
vg = −
=
∂f2 (x,t) / ∂x dk
Transverse magnetic (E)
Lecture 8 - E. Wilson - 3/20/2008 - Slide 9
modes
Transit time factor
z
Ez
-G/2
G/2
z
ωG ⎞
⎛
sin
⎜ 2 βc ⎟
+G 2
⎟ cos ϕ = E0 G (Γ ) cos ϕ
V = ∫ Eo cos(ωt + φ )dz = E0 G ⎜
ω
−G 2
⎜
⎟
⎝ 2βc ⎠
ωG ⎞
⎛
sin
⎜
+G 2
2 βc ⎟
V = ∫−G 2 Eo cos(ωt + φ )dz = E0 G ⎜ ω ⎟ cos ϕ = E0 G (Γ ) cos ϕ
⎜
⎟
⎝ 2β c ⎠
sin θ / 2
Γ=
.
θ /2
Lecture 8 - E. Wilson - 3/20/2008 - Slide 10
Cavity resonators
⎧∇ ⋅ E = 0 ; ∇ ⋅ H = 0
⎪
⎨
∂H
∂E
; ∇ × H = σE + ε
⎪⎩∇ × E = − μ
∂t
∂t
∂E
∂ 2E
∇ E = μσ
+ εμ 2
∂t
∂t
2
‹•
n x E= 0 – because the E field should be
normal to the perfectly conducting walls.
‹ Assume we can separate out a time
dependent solutions
aM = e
‹ leaving
−
ωM
2Q
t
{A1 cos Ω M t + A2 sin Ω M t}
a spatial
solution:
2
⎧∇ E + Λ 2 E = 0
⎪
⎨∇ ⋅ E = 0
⎪ η × E = 0 on the boundary
⎩
Lecture 8 - E. Wilson - 3/20/2008 - Slide 11
The cylindrical cavity
2
2
∂
ξ
∂ξ
∂
ξ
1
2
+ 2
∇ ξ = 2 +
∂r
r ∂r ∂z
‹ Ez
= F(r) · j (z) ; Er = Y(r) · f(z)
⎧
⎛ P0l ⎞
mπ
E
cos
=
E
J
r
z
⎟
⎪ z
0 0⎜
h
⎝ r0 ⎠
⎪
⎪⎪
mπ
mπ r0 ⎛ P0l ⎞
J1 ⎜
r⎟ sin
z
⎨ Er = E0
P0l h ⎝ r0 ⎠
h
⎪
⎪
2
⎛
⎞
P
m
π
⎞
⎛
⎪ Λ2 = ⎜ 0l ⎟ + ⎜
⎟
0lm
⎪⎩
⎝ r0 ⎠ ⎝ h ⎠
Lecture 8 - E. Wilson - 3/20/2008 - Slide 12
Lecture11 – Accelrating Cavities Icontents
‹ Necessary
conditions for acceleration
‹ Waves in free space
‹ Two travelling waves in a guide.
‹ A transverse electric (H) mode
‹ Phase velocity and Group velocity
‹ Transverse magnetic modes
‹ Transit time factor
‹ The cylindrical cavity
Lecture 8 - E. Wilson - 3/20/2008 - Slide 13