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 (projection)
along Q optical axis
4 magnetic poles, focusing element, forces
increase with r.
M axw ell equations for B field
  B  0
  B  V , V  0
B  0 
Solutio ns : V ( s , x , z )  G ( s )  x  z
Bx 
V
x
 G (s)  z
Bz  G (s)  x
E quations of m otion
m s  0 since p  m v  m s  const
mx  q  s  Bz   q  s  G (s)  x
mz   q  s  B x  q  s  G (s )  z
opposite signs
Ideal Thin Quadrupole Lens
2
incoming
particle
d x
ds

x(s=0)
0
f x< 0
s
dx
fx  
L: Lens
thickness
qG L
  fz
s

ds
Slope is proportional to x ( s  0)
p
2
x
 x  
q
p

q

2
L 2
L 2

q  G (s)
x
p
ds G (s)  x(s)
 G (0 )  x (0 )  L  tan 
p
 th in len s a p p ro xim a tio n
focus and defocus in x and z
focal length
Convex lens: fx > 0
Concave lens: fx < 0
Set fx =|fx|
q·G
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,..) and
directions (x’=dx/ds,…) of drift.
 x(s) 
u (s)  


s

x
(
s
)


u 2 ( s )   ( s , f , ..)  u 1 ( s )
E xam ple : thin quadrupole
x 2  x1
 fx
 | f x |
x 2  x 1  x 1 f x is represented by
  s, f x   Q f
focusing in x
 1

 s fx
0

1
 1
Qd  
 s fx
defocusing in x
0

1
Free Drift
T ranslation
s s 

1 
1
D r (s )  
0
x
 x1 
1
u1  
  u 2  D r (  s ) u1  
0
 x1s 
 s s   x1   x1  x1 s 

  u2

x1s
1   x1s  

Drift spaces make
visible the effect of
active beam elements
see MathCad
BeamTransp program
u1 = (x,0)  x’ = 0, no divergence
u2 = Dr(550cm)u1
s
Quadrupole Focusing Action
0
 x1 

  u1  

x
s
1
 1 
F ocusing w ith focal length f
 1
Q f (s , f )  
 s f
 1
u1  u 2  Q f ( s , f ) u1  
 s f
x1
0   x1  

  u2


1   x1s     s f  x1  x1s 
x
The larger x, the more
negative the slope.
Defocus: f  -f
u1 = (x,0)  x’=0, parallel to
nominal trajectory
u2 = Dr(100cm)u1
u3 = Dr(250)Qf(100)u2
s
Quadrupole Defocusing Action
D efocusing w ith focal length f
 1
u1  u 2  Q d ( s , f ) u1  
s f
 1
Q d (s , f )  
s f
0
 x1 

  u1  

x
s
1
 1 
x1
0   x1  

  u2


1   x1s    s f  x1  x1s 
x
The larger x, the more
negative the slope.
Defocus: f  -f
Quadrupole pairs: Qf·Qd
u1 = (x,0)  parallel entry
u2 = Dr(100cm)u1
s
u3 = Dr(250)Qd(100)u2
Sector Magnet
s
y
z
r
z
B eta tro n
s

x
o scilla tio n
z zz  0
2
 z  n 0
2
L
" W ave vector " k   z s
 z t  k s
z ( s )  z 0 cos ks  ( z 0 k ) sin ks
0 
Matrix representation of bending magnet in z-direction:
(1 ks ) sin kL 
;
cos kL

qB
m
tan   z 0
s  z ( s )   ksz 0 sin ks  sz 0 cos ks
 cos kL
BM  
  ks sin kL
2
 z 
u (s)  


sz


Bending Magnet
B ending
 cos kL
BM  
  ks sin kL
sin kL ks 

cos kL 
 z1 cos kL   z  k  sin kL 
 z1 
u1  

  u 2  B M u1  


z
s
 1 
  ksz1 sin kL  sz 1 cos kL 
z
N
S
u1 = (z,0), z’=0, parallel
u2 = Dr(100cm)u1
u3 = Dr(200)BM.u2
u3 = Dr(250)Qf(100)u2
s
Using Beam Transport Elements in Mass Spectroscopy
Used with ISOL target to measure exotic reaction product
new nuclides are
nuclei
produced in high
DRAGON
charge states 
(TRIUMF/Canada)
good mass resolution
12
Combination
Wien Filter - TOF
2 sets of magnetic and electric dipoles,
MCP TOF system in focal plane
L o re n tz fo rce

F  q v B  E

m a tch e d E B  f (v )
 W ie n F ilte r
B: r 
p
qB

mv
qB
Nuclear Masses
TO F :  v , m, K
Princ. accuracy
Abs :
Rel :
W. Udo Schröder, 2004
m
m
m
m
 10
5
 10
8