In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line.
If not specified further, the term is likely to refer to the longitudinal effective acceleration voltage
The acceleration voltage is an important quantity for the design of microwave cavities for particle accelerators.
For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path.
The following considerations are generalized for time-dependent fields.
The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity
along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge,[2]
For resonant structures, e.g. SRF cavities, this may be expressed as a Fourier integral, because the fields
, and the resulting Lorentz force
Since the particles kinetic energy can only be changed by electric fields, this reduces to
This is advantageous, since the relative phase between particle and the experienced field was fixed in the previous considerations (the particle travelling through
experienced maximum electric force).
To account for this degree of freedom, an additional phase factor
is included in the eigenmode field definition
which leads to a modified expression
In comparison to the former expression, only a phase factor with unit length occurs.
Thus, the absolute value of the complex quantity
is independent of the particle-to-eigenmode phase
It represents the maximum achievable voltage that is experienced by a particle with optimal phase to the applied field, and is the relevant physical quantity.
A quantity named transit time factor[2]
is often defined which relates the effective acceleration voltage
to the time-independent acceleration voltage
In this notation, the effective acceleration voltage
In symbolic analogy to the longitudinal voltage, one can define effective voltages in two orthogonal directions
that are transversal to the particle trajectory
which describe the integrated forces that deflect the particle from its design path.
Since the modes that deflect particles may have arbitrary polarizations, the transverse effective voltage may be defined using polar notation by
The tilde-marked variables are not absolute values, as one might expect, but can have positive or negative sign, to enable a range
Note that this transverse voltage does not necessarily relate to a real change in the particles energy, since magnetic fields are also able to deflect particles.
Also, this is an approximation for small-angle deflection of the particle, where the particles trajectory through the field can still be approximated by a straight line.