Transcript Slide 1

Beam Transport
Elaborate system of
vacuum beam lines
transports chargedparticle beam from
accelerator to
experimental site
Beam optics: mainly magnetic elements, function like lenses and
prisms for optical beams, change propagation direction, focus.
x
x
x
x
x
Effect of beam elements on phase
space profile x, y, z; x, y, z
x
shape of beam
profile in
phase space
can be
changed, not
the density
(Liouville’s
Theorem)
Active Beam Elements
•Dipole magnets change beam direction.
•Quadrupole magnets focus and defocus, used in pairs.
•Sextupole magnets make higher order focusing
corrections.
Modeling of beam line performance is done in matrix
formulation in phase space of charged particle.
Combinations of such elements are used for beam
transport to experimental sites and as magnetic
analyzers/filters/spectrometers.
Magnetic Quadrupole Lens
z
x
q: charge
s: average particle
path
4 magnetic poles, focusing element, forces
increase with r.
Maxwell equations for B field
  B  0
  B  V , V  0
B  0 
Solutions : V ( s, x, z )  G ( s )  x  z
V
Bx 
 G (s)  z
Bz  G ( s )  x
x
Equations of motion
ms  0 since p  mv  ms  const
mx  q  s  Bz  q  s  G (s )  x mz  q  s  Bx  q  s  G (s )  z
Ideal Thin Quadrupole Lens
incoming
f x< 0
d 2x
x q  G (s)

x  2 
x
2
ds
s
p

dx q  L 2
   ds G ( s )  x( s )
x(s=0)
ds p  L 2
q
  G (0)  x(0)  L  tan 
L: Lens
p
thickness
Slope is proportional to x (s  0)
p
fx  
  f z focal length
qGL
Convex lens: fx > 0
Concave lens: fx < 0
 thin lens approximation
focus and defocus in x and z
qG
focus defocus
0
z
x
<0
x
z
Matrix Formulation of Beam Transport
Approximation here: small x, z, decoupled x-z motion
Phase space trajectory made up of points
 x( s ) 
u (s)  
u2 ( s )  ( s, f ,..)  u1 ( s )

 s  x( s ) 
Example : thin quadrupole
x2  x1
x2  x1  x1 f x is represented by
 1
  s, f x   Q f  
 s f x
focusing in x
0
1 
 1
Qd  
 s fx
defocusing in x
0
1 
Free Drift
 1 s s 
Translation Dr (s )  

1 
0
 x1 
 1 s s   x1   x1  x1s 
u1     u2  Dr (s )u1  

 u2




1   x1s   x1s 
0
 x1s 
x
Drift spaces make
visible the effect of
active beam elements
see Mathcad
BeamTransp program
u1 = (x,0)  x’ = 0, no divergence
u2 = Dr(100cm)u1
z
Quadrupole Focusing Action
0
 x1 
 1
Focusing with focal length f Q f (s, f )  
  u1   xs 
 s f 1 
 1 
x1
0   x1  

 1
u1  u2  Q f ( s, f )u1  
  xs     s f x  xs   u2
 1 1 
  s f 1  1   
x
The larger x, the more
negative the slope.
Defocus: f  -f
u1 = (x,0)  parallel to nominal
trajectory
u2 = Dr(100cm)u1
u3 = Dr(250)Qf(100)u2
z
Quadrupole Defocusing Action
 x1 
 1 0
Defocusing with focal length f Qd (s, f )  
  u1   xs 
 s f 1
 1 
x1

 1 0   x1  
u1  u2  Qd ( s, f )u1  
  xs    s f x  xs   u2
 1 1 
 s f 1  1   
x
The larger x, the more
negative the slope.
Defocus: f  -f
u1 = (x,0)  parallel entry
u2 = Dr(100cm)u1
u3 = Dr(250)Qd(100)u2
z
Sector Magnet
s
y
r
z
s

x
L
Wave vector k   z s  z t  k s
z ( s)  z0 cos ks  ( z0 k )sin ks
s  z( s)  ksz0 sin ks  sz0 cos ks
Betatron
oscillation
z   z2 z  0
 z2  n 02
qB
0 
m
tan   z0
Matrix representation of bending magnet in z-direction:
(1 ks)sin kL 
 z 
 cos kL
BM  
; u ( s)   

cos kL 
 ks sin kL
 sz 
Bending Magnet
sin kL ks 
 cos kL
Bending BM  


ks
sin
kL
cos
kL


 z1 cos kL   x k  sin kL 
 x1 
u1     u2  BMu1  



x
s
 1 
 ksz1 sin kL  sx1 cos kL 
x
N
S
u1 = (x,0)
u2 = Dr(100cm)u1
u3 = Dr(200)BM.u2
u3 = Dr(250)Qf(100)u2
z