12
An Introduction to Beam Physics
FIGURE 1.7: The general principle of the Cockcroft-Walton generator.
(From J. D. Cockcroft and E. T. S. Walton, Proc. Royal Soc. London, A,
136:619, 1932 [14]. With permission.)
where r(t) is the particle’s position as a function of time and v(t) its velocity.
In the special case that
E is time independent and hence can be written
in terms of a potential via
E = −
∇V, this path integral reduces in a natural
way to the difference in potential as
K = q · (V ( r 1 ) − V ( r 2 )) .
This simple fact implies a very important consequence for the design of electric
accelerating fields: if there is to be any chance to utilize the same electric field
repeatedly for the purpose of acceleration, then the electric field has to
be time dependent, because otherwise repeated passing just results in a
periodic increase and decrease of energy. In fact, the attempt to build an
accelerator trying to increase energy repeatedly by flying through the same
time independent field is tantamount to the attempt to build a perpetual
motion machine.
1.3.1 Electrostatic Accelerators
The first class of important accelerators are those based on static electric
fields. The kind that grew directly out of the area of electric circuit is the
voltage multiplier that is now known under the name Cockcroft-Walton
generator. It consists of a simple but clever circuit made of diodes and capacitors, forming the voltage multiplier ladder. Each time a base voltage is
applied, the first capacitor is filled with charge. Each time the voltage is removed, the first capacitor passes on some of its charge to the second capacitor,
which in turn feeds the third capacitor, and so on. Depending on the number
An Introduction to Beam Physics
FIGURE 1.7: The general principle of the Cockcroft-Walton generator.
(From J. D. Cockcroft and E. T. S. Walton, Proc. Royal Soc. London, A,
136:619, 1932 [14]. With permission.)
where r(t) is the particle’s position as a function of time and v(t) its velocity.
In the special case that
E is time independent and hence can be written
in terms of a potential via
E = −
∇V, this path integral reduces in a natural
way to the difference in potential as
K = q · (V ( r 1 ) − V ( r 2 )) .
This simple fact implies a very important consequence for the design of electric
accelerating fields: if there is to be any chance to utilize the same electric field
repeatedly for the purpose of acceleration, then the electric field has to
be time dependent, because otherwise repeated passing just results in a
periodic increase and decrease of energy. In fact, the attempt to build an
accelerator trying to increase energy repeatedly by flying through the same
time independent field is tantamount to the attempt to build a perpetual
motion machine.
1.3.1 Electrostatic Accelerators
The first class of important accelerators are those based on static electric
fields. The kind that grew directly out of the area of electric circuit is the
voltage multiplier that is now known under the name Cockcroft-Walton
generator. It consists of a simple but clever circuit made of diodes and capacitors, forming the voltage multiplier ladder. Each time a base voltage is
applied, the first capacitor is filled with charge. Each time the voltage is removed, the first capacitor passes on some of its charge to the second capacitor,
which in turn feeds the third capacitor, and so on. Depending on the number