The tune shift with amplitude is an important concept in circular
accelerators or
synchrotron
A synchrotron is a particular type of cyclic particle accelerator, descended from the cyclotron, in which the accelerating particle beam travels around a fixed closed-loop path. The strength of the magnetic field which bends the particle beam i ...
s. The machine may be described via a symplectic one turn map at each position, which may be thought of as the Poincaire section of the dynamics.
A simple harmonic oscillator has a constant tune for all initial positions in phase space. Adding some non-linearity results in a variation of the tune with amplitude.
Amplitude may refer to either the initial position, or more formally, the initial action of the particle.
Definition
Consider dynamics in
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. These dynamics are assumed to be determined by a Hamiltonian, or a
symplectic map. For each initial position, we follow the particle as it traces out its orbit. After transformation into
action-angle coordinates
In classical mechanics, action-angle variables are a set of canonical coordinates that are useful in characterizing the nature of commuting flows in integrable systems when the conserved energy level set is compact, and the commuting flows are com ...
, one compute the tune
and the
action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
. The tune shift with amplitude is then given by
. The transformation to action-angle variables out of which the tune may be derived may be considered as a transformation to
normal form.
Significance
The tune shift with amplitude is important as a measure of non-linearity of a system. A linear system will have no tune shift with amplitude. Further, it can be important regarding the stability of the system. When the tune reaches resonant values, it can be unstable, and thus a tune-shift with amplitude can limit the
stability region, or
dynamic aperture.
Examples of systems with tune shift with amplitude
In
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, a simple example of a system with tune shift with amplitude is a
pendulum
A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate i ...
. In accelerator physics, both the transverse and the longitudinal dynamics show tune shift with amplitude. A simple model of the transverse dynamics is of an oscillator with a single
sextupole, it is referred to as the
Hénon map
In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaos theory, chaotic behavior. The Hénon map takes ...
. Another model for this case is the
Standard Map
The standard map (also known as the Chirikov–Taylor map or as the Chirikov standard map) is an area-preserving chaotic map from a square with side 2\pi onto itself. It is constructed by a Poincaré's surface of section of the kicked rot ...
.
An important example is the typical case of distributed sextupoles in a storage ring.
Computation
The tune shift with amplitude may be computed in numerous ways. One involves the use of the normal form method. See for the use of this method for the pendulum.
It may also be computed by tracking the orbit through phase space, and then Fourier transforming the projections onto the different planes. For computation in the Elegant code, see
The tune may also be computed by a refinement over the Fourier transform method, called NAFF. e.g.
sddsNAFF
It may also be computed analytically via a formula, using the normal form method, otherwise. For the storage ring case with distributed sextupoles, one can see [J. Bengtsson, "The Sextupole Scheme for the Swiss Light Source (SLS): An Analytic Approach," SLS Note 9/97, March 7, 1997.]
See also
anharmonicity
In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmo ...
References
{{Reflist
Accelerator physics