Beams and Beam Physics
11
FIGURE 1.6: Layout of the first electron cyclotron resonance (ECR) ion
source that produced multiple charged ions. (Reprinted with permission from
R. Geller, Appl. Phys. Lett., 16:401, 1970 [28]. Copyright 1970, AIP Publishing LLC.)
of the relative motion, often a linear approximation with all the resulting
simplifications is possible, but frequently a full understanding of the motion
can only be achieved by considering the nonlinear effects.
Considering the special shape of the Lorentz force law (see eq. (1.1)),
since v ×
B is perpendicular to the velocity v, it is apparent that magnetic
fields cannot be used for purposes of acceleration, which requires forces in
the direction of the particle. Thus any acceleration has to be provided by
electric fields. However, as we shall see, also magnetic fields have very good
use in particle accelerators, as they can be employed to guide the beam to
where it is needed. In particular, in the process of acceleration they are often
used to guide the beam through the same region of electric field repeatedly
and thus allow the device to maximize the use of the electric fields. Indeed,
for this purpose of guiding the beam, magnetic fields are usually even better
suited than electric fields. This is because for the high velocities that beams
usually have after even modest acceleration, the forces that can be attained
with technologically available magnetic fields far exceed those that can be
achieved with the respective electric fields.
Very generally, the amount of energy K a particle gains while traveling
from time t 1 to time t 2 in an electric field
E( r, t) that depends on position
and time is given by the path integral
K = q ·
t2
t1
E( r(t), t) · v(t) dt,
11
FIGURE 1.6: Layout of the first electron cyclotron resonance (ECR) ion
source that produced multiple charged ions. (Reprinted with permission from
R. Geller, Appl. Phys. Lett., 16:401, 1970 [28]. Copyright 1970, AIP Publishing LLC.)
of the relative motion, often a linear approximation with all the resulting
simplifications is possible, but frequently a full understanding of the motion
can only be achieved by considering the nonlinear effects.
Considering the special shape of the Lorentz force law (see eq. (1.1)),
since v ×
B is perpendicular to the velocity v, it is apparent that magnetic
fields cannot be used for purposes of acceleration, which requires forces in
the direction of the particle. Thus any acceleration has to be provided by
electric fields. However, as we shall see, also magnetic fields have very good
use in particle accelerators, as they can be employed to guide the beam to
where it is needed. In particular, in the process of acceleration they are often
used to guide the beam through the same region of electric field repeatedly
and thus allow the device to maximize the use of the electric fields. Indeed,
for this purpose of guiding the beam, magnetic fields are usually even better
suited than electric fields. This is because for the high velocities that beams
usually have after even modest acceleration, the forces that can be attained
with technologically available magnetic fields far exceed those that can be
achieved with the respective electric fields.
Very generally, the amount of energy K a particle gains while traveling
from time t 1 to time t 2 in an electric field
E( r, t) that depends on position
and time is given by the path integral
K = q ·
t2
t1
E( r(t), t) · v(t) dt,