Alexej Grudiev
2013/10/23
ICE section meeting

• Many thanks to Elias Metral and Alexey Burov for listening to me and explaining what actually Landau damping is !

• 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: 𝑄 𝑥
= 𝑄
0
+ 𝑎𝐽 𝑥
+ 𝑏𝐽 𝑦 𝜔 𝑥 𝜔 𝑦 𝜔 𝑧
= 𝜔
𝐿𝑥 𝜔
𝐿𝑦 𝜔
𝐿𝑧
+ 𝑎 𝑥𝑥 𝑎 𝑦𝑥 𝑎 𝑧𝑥 𝑎 𝑥𝑦 𝑎 𝑦𝑦 𝑎 𝑧𝑦 𝑎 𝑥𝑧 𝑎 𝑦𝑧 𝑎 𝑧𝑧
𝐽
𝐽
𝐽 𝑥 𝑦 𝑧
/𝜖
/𝜖
/𝜖 𝑥 𝑦 𝑧
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 v y and v x 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

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
ΔQ coh
=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

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:
And for an RF quadrupole: n=2
E kick
F



E
 e

E kick
 e
 j
 c v z z
;
 p

( r ,

)

L

0
F
 v z dz

H
B kick kick


 v z v z
 c e c
L

0
 c
H
 e


E e kick j
 c

E kick
 z

Z
0 u
Z
0 u z
 z

H kick
H kick
 dz
 p

( r ,

) p

( 2 )
( r ,

)

 n



0
1 c p

( n ) ( r ,

) r u
 r cos( 2

)
 u  sin( 2

)
L

0

F

( 2 ) dz
For ultra-relativistic particle, equating the
RF and magnetic kicks, RF quadrupolar strength can be expressed in magnetic units:
B
( 2 ) 
1 ec
F

( 2 ) b
( 2 )  
L
0
B
( 2 ) dz
[ T / m ]
[ Tm / m ]
Strength of RF quadruple B’L depends on RF phase:
(for bunch centre: r=ϕ=z=0)
B ' L ( s )
  

 b ( 2 ) e j
 c z




Accelerating Voltage:
V acc
( r ,

)

L

0 dzE acc
( r ,

, z )

L

0 dzE z
( r ,

, z )
 e j
 c z
Can also be expanded in terms of multipoles:
Panofsky-Wenzel (PW) theorem:
Gives an expression for quadrupolar RF kicks:
V acc
( r ,

)
  n r n cos( n

) V ( acc n ) p

( r ,

)
 where :



 je

 u r

L
0

 r dz



E acc
( r ,

,

 u

1 r

  z ) ;

  n r n cos( n

)
L

0 dzE ( n ) acc for E ~ e
 j
 t p
(

2 )
( r ,

)
 je

2 r
  u r cos( 2

)

 u
 sin( 2

)


L
0
E
( 2 ) acc
( z ) dz
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)
2
 j
( 2 )
E acc

B
( 2 )
2
 j
( 2
V acc
)
[ T / m ]

2
 j 
L
0
( 2 )
E acc dz
 b
( 2 )
[ Tm / m ]
V acc
( r ,

, z )
 



(
V acc
2 ) e j
 c z


 r
2 cos( 2

)

Main RF (φ s
= 0 at zero crossing): V acc
( z )

V
0 sin h

0 c z
  s 0


;

2 s 0
 
0
2
 hV
0 cos(
 s
2
 c

B
0
0
)
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 (φ s2
= 0 ):
Synchrotron frequency for
Main RF + RF quad voltage:
Q
V acc
( r ,

, z )
 ( 2 ) jV acc sin

 c




( 2
V acc
) e j
 c z



 r
2 z

( y
2  x
2
) cos(

V
2 sin
2

 c
)
 s
2  
0
2
 hV
0 cos(
 s
2
 c

B
0
0
)

 1
 h
2
V
2 cos(
 hV
0 cos( s

2 s

0
)
0 )

 ;
 s
  s 0
1
 hV
0 h
2
V
2 cos(
 s 0
) z
V
2

 b
( 2 )
2
( y
2  x
2
)
  s 0

 1

1
2 hV
0 h
2
V
2 cos(
 s 0
)

   s 0



1

1
2

0
2
 s
2
0
 h
2
V
2
2
 c

B
0



Useful relation for stationary bucket: 
2 s 0

 h

0 eV
0 c
2

E
;

E
2 
2

EeV
0
 h
bucket hight

 s
E
ˆ
0

 h

0
;
2 E

E
  z

E
ˆ

  z

 s 0
E 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: 𝜔 𝜔 𝜔 𝑥 𝑦 𝑧
= 𝜔
𝐿𝑥 𝜔
𝐿𝑦 𝜔
𝐿𝑧
+ 𝑎 𝑥𝑥 𝑎 𝑦𝑥 𝑎 𝑧𝑥 𝑎 𝑥𝑦 𝑎 𝑦𝑦 𝑎 𝑧𝑦 𝑎 𝑎 𝑎 𝑥𝑧 𝑦𝑧 𝑧𝑧
𝐽
𝐽
𝐽 𝑥 𝑦 𝑧
/𝜖
/𝜖
/𝜖 𝑥 𝑦 𝑧
Both horizontal and vertical transverse spreads of synchrotron tune are non-zero for RF quadrupole
 s
  s 0 x , y


1
2


0
2 s 0
 h
2
V
2
2
 c

B
0
2 J x , y
 x , y
  s 0


8

0 cos(
 x , y
); x , y b
( 2 )

B
0
 x , y
 c
2

 s 0 c
( y 2

J x , y
 x , y
 x 2 )
  s


8

0 b ( 2 )

B
0
 c
2 
 c s 0
( J y
 y

J x
 x
) a zx

If

  x
 x z

E
/

8

0
E
  b ( 2 )

B
0
 c
2 
 c s 0
; a zy
   y x , y matrix is symmetric
 y

8

0
: a zx
 b ( 2 )

B
0
 c a xz
; a zy
2 
 s 0
 a yz c
Longitudinal spread of both horizontal and vertical betatron tunes is non-zero for RF quadrupole
B ' L ( z )
 b ( 2 ) cos


 c z
 b ( 2 ) 



1

 c
2 z 2
2
 o ( z 4

)


 z

2 J z
 z cos(
 z
);
 z
2
 z

J z
 z
  z
2

J z z

Q x , y
  x , y


 x , y
4

KL

  x , y

4

0
 x , y b ( 2 )
4

B
0

( s )




1

B ' L

B
0
 c a xz
   x

8

0 b
( 2 )

B
0


2
 z
2 c
2  z
J z
2


 focusing a yz
   y

8

0 b ( 2 )

B
0
 c
2
 z
2 de focusing
7um >> 0.5nm, for LHC 7TeV => a zx
↑TDR longitudinal ε z
<< a xz
; a zy
<< a yz
=> Longitudinal spread is much more effective
(4σ) =2.5eVs and transverse normalized ε N x,y
(1σ)=3.75um emittances are used

Longitudinal spread of the betatron tune a yz a yz
• In the case of RF quadrupole a yz is not zero. One can just substitute a yz
• Make longitudinal tune spread a yz instead of ma zz larger than the coherent tune shift (ΔΩ m
) -> Landau damping.
• The same is true for horizontal plane. This gives the RF quadrupole strength: a yz
  y

8

0 b
( 2 )

B
0
 c

2
 z
2    
Q y coh

0
 b ( 2 )  
B
0
2


2
 z
2

Q x, y coh
 x , y

b ( 2 )  
B
0
2


2
 z
2

Q x, y coh
 x , y
• |ΔQ coh
|≈2e-4
• β≈200m
• σ z
=0.08m
• λ=3/8m
• ρ=2804m
• B
0
=8.33T
-----------------------
• b (2) =0.33Tm/m

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] b (2) [Tm/m]
Max(Bsurf) [mT]
Max(Esurf) [MV/m]
1
0.0143
12
4.6

• 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 ΔQ coh
~2e-4 at 7TeV
• Maybe some tests can be done in SPS with one cavity prototype…
• More work needs to be done…