Transcript RF quadrupole for Landau damping (Alexej Grudiev)

RF quadrupole for
Landau damping
Alexej Grudiev
2013/10/23
ICE section meeting
Acknowledgements
• Many thanks to Elias Metral and Alexey Burov
for listening to me and explaining what
actually Landau damping is !
Outline
• Introduction
– Reminder (many for myself) of what is a stability
diagram for Landau damping
– Parameters of the LHC octupole scheme for Landau
damping
• Longitudinal spread of the betatron tune induced
by an RF quadrupole
• Transverse spread of the synchrotron tune
induced by an RF quadrupole
• Parameters of Landau damping scheme based on
RF quadrupole
Stability diagrams for Landau damping (1)
Berg, J.S.; Ruggiero, F., LHC Project Report 121, 1997
Stability diagrams for Landau damping (2)
Berg, J.S.; Ruggiero, F., LHC Project Report 121, 1997
Explicit form for 3D-tune linearized in terms of
action:
Octupole tune spread: 𝑄𝑥 = 𝑄0 + 𝑎𝐽𝑥 + 𝑏𝐽𝑦
𝜔𝑥
𝜔𝐿𝑥
𝜔𝑦 = 𝜔𝐿𝑦
𝜔𝑧
𝜔𝐿𝑧
𝑎𝑥𝑥
+ 𝑎𝑦𝑥
𝑎𝑧𝑥
𝑎𝑥𝑦
𝑎𝑦𝑦
𝑎𝑧𝑦
𝑎𝑥𝑧
𝑎𝑦𝑧
𝑎𝑧𝑧
𝐽𝑥 /𝜖𝑥
𝐽𝑦 /𝜖𝑦
𝐽𝑧 /𝜖𝑧
Potential well distortion,
actually non-linear !
Stability diagrams for Landau damping (3)
Berg, J.S.; B Ruggiero, F., LHC Project Report 121, 1997
• 2D tune spread is more effective if vy
and vx have opposite signs
• It is done by means of octupoles
• Make transverse tune spread larger
than the coherent tune shift -> Landau
damping
Stability diagrams for Landau damping (4)
Berg, J.S.; B Ruggiero, F., LHC Project Report 121, 1997
• Transverse oscillation stability curves for longitudinal tune spread are qualitatively
similar to the ones for 1D transverse tune spread
• Make longitudinal tune spread larger than the coherent tune shift -> Landau damping
LHC octupoles for Landau damping
Landau damping, dynamic aperture and octupoles in LHC, J. Gareyte, J.P. Koutchouk and F.
Ruggiero, LHC project report 91, 1997
80 octupoles of
0.328m each are
nesessary to
Landau damp
the most
unstable mode
at 7 TeV with
ΔQcoh=0.223e-3
In LHC, 144 of
these octupoles
(total active
length: 47 m)
are installed in
order to have
80% margin and
avoid relying
completely on
2D damping
What is it, an RF quadrupole ?
For given EM fields
Lorenz Force (LF):
Gives an expression for kick
directly from the RF EM fields:
Which can be expanded in
terms of multipoles:
j
E kick  E   e

H kick  H   e
;
F   e  E kick  v z  B kick 
L
p (r , ) 
F

dz 
vz
0
vz c
e
vz c
c
j

z
c

e E kick  Z 0 u z  H kick
 E

L
kick
p (r , ) 

p  (r , ) 
(2)
For ultra-relativistic particle, equating the
RF and magnetic kicks, RF quadrupolar
strength can be expressed in magnetic
units:
1
0
p  (r , )
(n)

B
 F
L
r u r cos( 2 )  u  sin( 2 )
c
(2)

0
1

(2)
ec
F
(2)
[T / m ]
(2)
dz
[Tm / m ]
L
b
(2)

Strength of RF quadruple B’L depends on RF phase:
(for bunch centre: r=ϕ=z=0)
B

 Z 0 u z  H kick dz

n0
And for an RF quadrupole: n=2
z
c
0
 (2) j z 
B ' L ( s )   b e c 


dz
Appling Panofsky-Wenzel theorem to
an RF quadrupole
L
V acc ( r ,  ) 
Accelerating Voltage:
L
 dzE
acc
(r , , z ) 
0
 dzE z ( r ,  , z )  e
j

z
c
0
L
Can also be expanded in terms of multipoles:
V acc ( r ,  ) 
r
n
cos( n  )V
(n)
acc

n
p (r , ) 
Panofsky-Wenzel (PW) theorem:
n
L
 je
 dz 

r
n

E acc ( r ,  , z ) ;
cos( n  )  dzE
0
for
E ~e
 j t
0
 
 1 
where :    u r
 u
r
r 
p (r , ) 
Gives an expression for quadrupolar RF kicks:
(2)
2j
For ultra-relativistic particle, equating the RF
and magnetic kicks, accelerating voltage
quadrupolar strength can be calculated from
magnetic quadrupolar strength:
Accelerating voltage of RF quadruple:
(for bunch centre r=ϕ=z=0)

2j

V acc

E acc  B
(2)
V
(2)
acc

L


(2)
2 r u r cos( 2 )  u  sin( 2 )  E acc ( z ) dz
je
0
(2)
2j

[T / m ]
L

E acc dz  b
(2)
(2)
[Tm / m ]
0
 (2) j z  2
( r ,  , z )   V acc e c  r cos( 2 )


(n)
acc
Synhrotron frequency in the presence
of an RF quadrupole
 h 0

V acc ( z )  V 0 sin 
z   s 0 ;
 c

Main RF (φs = 0 at zero crossing):
RF quad voltage, if b(0) is real.
The centre of the bunch is on crest
for quadrupolar focusing but it is
on zero crossing for quadrupolar
acceleration (φs2 = 0 ):
Synchrotron frequency for
Main RF + RF quad voltage:
 
2
s
V
Q
acc
 s0   0
2
2
 hV 0 cos(  s 0 )
2 c  B 0
 (2) j z  2
( r ,  , z )   V acc e c   r cos( 2 )


 
 
(2)
2
2
 jV acc sin  z   ( y  x )  V 2 sin  z 
 c 
 c 
2
0
 hV 0 cos(  s 0 ) 
 s   s0
Useful relation for stationary bucket:
h 2V 2 cos(  s 2  0 ) 
1


;
hV
cos(

)
0
s0


2 c  B 0
V2 
b
(2)
(y  x )
2
2
2
2



1
h 2V 2
1  0  h 2V 2 
1
  s 0 1 

   s 0 1 
2
hV 0 cos(  s 0 )
2
hV
cos(

)
2

2

c

B
0
s0 
s
0
0



h 2V 2
 s0 
2
 s0
 Eˆ

 h  0 eV 0 c
2  E
 h 0
2E
;
;
2 EeV 0
2
 Eˆ 
- bucket hight
h
E z
 Eˆ


 s0 E
z
 hc
Longitudinal spread of betatron tune and
Transverse spread of synchrotron tune induced
by RF quadrupole
Explicit form for 3D-tune linearized in terms of action:
𝑎𝑥𝑥
+ 𝑎𝑦𝑥
𝑎𝑧𝑥
𝜔𝑥
𝜔𝐿𝑥
𝜔𝑦 = 𝜔𝐿𝑦
𝜔𝑧
𝜔𝐿𝑧
𝑎𝑥𝑦
𝑎𝑦𝑦
𝑎𝑧𝑦
𝑎𝑥𝑧
𝑎𝑦𝑧
𝑎𝑧𝑧
𝐽𝑥 /𝜖𝑥
𝐽𝑦 /𝜖𝑦
𝐽𝑧 /𝜖𝑧
B' L(z)  b
Both horizontal and vertical transverse spreads of
synchrotron tune are non-zero for RF quadrupole
 s   s0 
x, y 
1   h 2V 2
2
0
2  s 0 2 c  B 0
  s0 
2 J x , y  x , y cos(  x , y );
   c
2
2
(y  x )
 
8  B 0  c   s 0
0 b
x, y
(2)
 x ,y
2
 J x,y  x,y
a zx
a xz
(2)
0 b     c
   c
   x x
; a zy    y  y
 
 
8  B 0  c   s 0
8  B 0  c   s 0
(2)
If  z  E / E   x , y
2
2
a yz
   2 z 2

 
(2)
4
cos  z   b 1   
 o( z )
 c 
  c  2

2 J z  z cos(  z );
Q x,y 
2
(2)
0 b
z
(2)
 x , y KL
4
 x,y    x,y
   c
 s 
(J y y  J x x )
 
8  B 0  c   s 0
0 b
Longitudinal spread of both horizontal
and vertical betatron tunes is non-zero
for RF quadrupole

 z
2
 J z z   z
2
z
4
B0
   2  J
z z
1   
4  B 0    c 
2
(2)
0 b
(2)
z
 x,y (s) B ' L
0 b
0 b
Jz
(2)



2
  2
  x
  z
8  B 0  c 
focusing
2
  2
  y
  z
8  B 0  c 
de - focusing
matrix is symmetric : a zx  a xz ; a zy  a yz
7um >> 0.5nm, for LHC 7TeV => azx << axz; azy << ayz => Longitudinal spread is much more effective
↑TDR longitudinal εz(4σ) =2.5eVs and transverse normalized εNx,y(1σ)=3.75um emittances are used
Longitudinal spread of the betatron tune
ayz
ayz
• In the case of RF quadrupole ayz is not zero. One can just substitute ayz instead of mazz
• Make longitudinal tune spread ayz larger than the coherent tune shift (ΔΩm) -> Landau
damping.
• The same is true for horizontal plane. This gives the RF quadrupole strength:
a yz
0 b
(2)
2
  2
y
 y
   z     Q coh  0
8  B 0  c 

b
(2)
 B0
2 
2
 z
2
 Q coh
x, y
 x,y
RF quadrupole in IR4
b
(2)
 B0
2 
2
 z
2
 Q coh
x, y
 x,y
• |ΔQcoh|≈2e-4
• β≈200m
• σz=0.08m
• λ=3/8m
• ρ=2804m
• B0=8.33T
----------------------• b(2)=0.33Tm/m
800 MHz Pillbox cavity RF quadrupole
H-field
E-field
For a SC cavity,
Max(Bsurf) should
nor be larger than
~100mT =>
One cavity can do:
b(2) =0.12 [Tm/m]
----------------------3 cavity is enough.
1 m long section
Stored energy [J]
1
b(2) [Tm/m]
0.0143
Max(Bsurf) [mT]
12
Max(Esurf) [MV/m]
4.6
Conclusion
• RF quadrupole can provide longitudinal
spread of betatron tune for Landau damping
• A ~1 m long cryomodule with three 800 MHz
superconducting pillbox cavities in IR4 can
provide enough tune spread for Landau
damping of a mode with ΔQcoh~2e-4 at 7TeV
• Maybe some tests can be done in SPS with
one cavity prototype…
• More work needs to be done…