In
accelerator physics
Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
, the term acceleration voltage means the effective
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
surpassed by a charged
particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
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 accelerator
A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams.
Large accelerators are used for fundamental research in particle ...
s. See also
shunt impedance
In accelerator physics, shunt impedance is a measure of the strength with which an eigenmode of a resonant radio frequency structure (e.g., in a microwave cavity) interacts with charged particles on a given straight line, typically along the axis ...
.
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.
Longitudinal voltage
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,
.
For resonant structures, e.g.
SRF cavities, this may be expressed as a
Fourier integral
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, because the fields
, and the resulting
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
, are proportional to
(
eigenmode
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s)
with
Since the particles
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
can only be changed by electric fields, this reduces to
Particle Phase considerations
Note that by the given definition,
is a
complex quantity. 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
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, an additional phase factor
is included in the
eigenmode
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
field definition
which leads to a modified expression
for the voltage. 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.
Transit time factor
A quantity named ''transit time factor''
is often defined which relates the effective acceleration voltage
to the time-independent acceleration voltage
.
In this notation, the effective acceleration voltage
is often expressed as
.
Transverse 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
with the ''polarization angle''
The tilde-marked variables are not absolute values, as one might expect, but can have positive or negative sign, to enable a range