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Figure 2.2:
Field lines in a magnetic quadrupole. For an ion moving in the sdirection, the force components
are focusing in the direction and defocusing in the direction.

In the present section we discuss focusing in quadrupole fields.
A magnetic quadrupole field is described by

(2.1) 
Such a field is produced by a magnet configuration with hyperbolic pole shapes, as shown in Fig. 2.2.
The equations of motion in a magnetic quadrupole are

(2.2) 
We eliminate the time and write the above equations as trajectory equations.
With
we get

(2.3) 
with the focusing strength for a magnetic quadrupole

(2.4) 
for electric quadrupole one obtains
We can write the solution in matrix form as
where and are the initial coordinates and and are the final values.
The matrix is called a transfer matrix. For
building beam transport lines we are interested in
quadrupole magnets that give transverse focusing
when and defocusing when .
Furthermore we have fieldfree drift spaces with .
Assuming force functions , which are piecewise constant inside the elements
and change steplike at the element boundaries,
we can obtain the following results for the corresponding transfer matrices
Focusing quadrupole () of length :

(2.5) 
Defocusing quadrupole () of length :

(2.6) 
Drift space () of length :

(2.7) 
The matrix for the passage through the whole transfer line is then
obtained by multiplication of the matrices of all the
transfer line elements
.
Next: Periodic Focusing Channel
Up: Linear Beam Dynamics without
Previous: Beam particle coordinates
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Oliver BoineFrankenheim
20010709