User:JsePrometheus/Longitudinal focusing

Phase-focusing or "longitudinal focusing", is a technique used in particle accelerators including synchrotrons, synchrocyclotrons, alternating current linear accelerators (linacs) and non-isochronous  fixed-field alternating gradient (FFAG) accelerators. Such particle accelerators use a device known as a radiofrequency (RF) accelerating cavity to apply a time-dependent electromotive force to accelerate a particle beam to higher  energies. Consequently the force on a given particle depends on its arrival time and this can be exploited to form a stable focusing force that keeps the particle on a synchronized trajectory.

History
Edwin McMillan and Vladimir Veksler are both credited with independent discoveries of the phase-focusing principle around 1945. Since that time, the longitudinal stability of particles has been a critical part of all high-energy particle accelerator design, excluding only isochronous cyclotrons.

The synchrotron oscillation was named after the synchrotron because that was the accelerator in which it was first discovered.

Synchrotron Oscillation
A synchrotron oscillation is an oscillation in the longitudinal features of a particle beam under the influence of a RF accelerating cavity in a cyclic accelerator. The longitudinal features of the beam are usually parameterized as a momentum coordinate $$\delta$$ and phase coordinate $$\phi$$. The momentum coordinate $$\delta$$ is the fractional deviation from a reference momentum $$p_{0}$$ given by $$\delta = \frac{p-p_{0}}{p_{0}}$$. The phase coordinate $$\phi$$ is the phase with respect to an RF cavity, which is given by $$\phi = \pi\frac{T_{rev} -hT_{rf}}{T_{rf}}$$ if the particle revolution period is $$T_{rev}$$ and a harmonic multiple of the rf period is $$hT_{rev}$$.

With these coordinates, the change due to the rf force in one revolution of a cyclic accelerator is given by the symplectic  map:

$$\Delta \delta = \frac{V}{\beta^{2} E} \sin\phi,~\Delta \phi = 2\pi h \eta \delta $$

where $$V$$ is the voltage across the accelerating gap of the rf cavity, $$\beta = v/c$$ is the particle velocity as a fraction of the speed of light,$$E$$ is the particle energy and $$\eta$$ is the phase-slip factor defined below.

Expressing the one-turn symplectic map given above as coupled first-order differential equations we have:

$$\dot{\delta} = f_{rev} \frac{V}{\beta^{2} E} \sin\phi,~\dot{\phi} = 2\pi h f_{rev} \eta \delta $$

The Hamiltonian of the coupled equations is given by:

$$H = \pi h f_{rev} \eta \delta^{2} + f_{rev} \frac{V}{\beta^{2} E} (1-\cos\phi)$$

These equations can be expressed as a single second-order differential equation given by:

$$\ddot{\phi} = 2\pi f_{rev}^{2} h \eta \frac{V}{\beta^{2} E} \sin\phi $$

By inspection, in can be seen that these equations are isomorphic to the  simple pendulum under the transformation $$\pi f_{rev} h \eta \rightarrow \frac{1}{2ml^{2}}$$ and $$f_{rev} V \rightarrow mgl$$. The frequency of small oscillations is given by $$\omega^{2} = 2\pi f_{rev}^{2} h |\eta| \frac{V}{\beta^{2} E}$$ and this is known as the synchrotron frequency. Typically a single synchrotron oscillation might occurs have hundreds of revolutions, making it an unusually low dynamical frequency compared to the RF frequency, the revolution frequency, and the betatron frequency.

In a synchrotron oscillation, the stable phase-space area is bound by a  separatrix. This region of phase space is commonly refer to as a bucket, because it has the effect of transporting particles.

Running Bucket
If the rf accelerating cavity of a cyclic accelerator remains at a fixed frequency, the particle beam will not undergo any net acceleration and instead it will simply maintain a synchronous momentum. By adiabatically changing the the frequency of the rf accelerating cavity (for instance, by adjusting the magnetic permeability with an applied magnetic field), the momentum of the particle beam will change to match the new synchronization conditions. In this way a particle beam can be accelerated to higher energies while maintaining synchronization with the rf accelerating cavity.

A linearly changing frequency can be expressed as change in phase per turn, known as the synchronous phase $$\phi_{s}$$. Including a nonzero synchronous phase, the linearized synchrotron equations become:

$$\Delta \delta = \frac{V}{\beta^{2} E} (\sin\phi - \sin\phi_{s}),~\Delta \phi = 2\pi h \eta \delta $$

$$\dot{\delta} = f_{rev} \frac{V}{\beta^{2} E} (\sin\phi - \sin\phi_{s}),~\dot{\phi} = 2\pi h f_{rev} \eta \delta $$

$$H = \pi h f_{rev} \eta \delta^{2} + f_{rev} \frac{V}{\beta^{2} E} (\cos\phi_{s}-\cos\phi + (\phi_{s}-\phi)\sin\phi_{s} )$$

$$\ddot{\phi} = 2\pi f_{rev}^{2} h \eta \frac{V}{\beta^{2} E} (\sin\phi - \sin\phi_{s}) $$

The greater $$\phi_{s}$$ is, the smaller the bucket area is. Therefore rapid acceleration will require greater beam quality or it will lead to greater particle loss.

Phase-slip Factor
The phase-slip factor $$ \eta $$ is a factor that describes how to relate changes in particle momentum to changes in the revolution period. Since this analysis is usually linearized, often the phase-slip factor simply refers to the first order dependence:

$$ \eta = -\frac{1}{f} \frac{\partial f}{\partial \delta} = \frac{1}{T} \frac{\partial T}{\partial \delta} = \left( \frac{1}{C} \frac{\partial C}{\partial \delta} - \frac{1}{v} \frac{\partial v}{\partial \delta} \right) $$

Changes in the revolution period from a change in momentum stem from two effects. Higher momentum particles have greater velocity, so the distance traveled by the particle will take less time. Note that as a particle approaches relativistic velocities, momentum makes a  diminishing contribution to velocity. Higher momentum particles also have greater beam rigidity, so the path length will change. In a typical accelerator (primarily composed of strong focusing cells and dipole magnets), the  dispersive effects will cause the higher momentum particles to travel a greater path length. Therefore the phase-slip factor can be positive or negative.

For a fixed reference orbit and focusing strength the contribution to the phase-slip factor made by changes in path length is constant. This is referred to as the momentum compaction factor $$ \alpha_{c} $$. Writing the phase-slip factor out in terms of the momentum compaction factor and relativistic parameters, we have:

$$ \eta = \alpha_{c} - \frac{1}{\gamma^{2}} $$

where $$ \gamma $$ is the Lorentz factor that shows up in the formulation of relativistic momentum. From this expression, it can be seen that the phase-slip factor $$\eta$$ will be zero for a particular value of $$\gamma$$, and the energy at which this occurs in known as the transition energy.

Below transition energy the phase-slip factor is negative and above transition energy the phase-slip factor is positive. When the phase-slip factor changes sign, the stable and unstable equilibrium points switch. Consequently phase-focusing can only be maintained by adjusting the phase of the RF cavity by $$ \pi $$ when accelerating through transition energy. When some but not all particles in a beam are above transition this is called negative mass instability because those particles behave as though they had negative mass. To minimize particle loss and reduction in beam quality, cyclic accelerators typically try to pass through transition energy and establish the new RF phase as rapidly as possible. The magnetic lattice may be tuned to change the transition energy simultaneously. Alternatively a cyclic accelerator may be designed so that it always operates either below or above the transition energy for that ring. With special lattice design, it may be possible to have a negative transition energy thereby avoiding the issue completely.

Isochronous Conditions
A cyclic accelerator is considered isochronous if the revolution period of the particle beam is independent of its momentum. Since the phase-slip factor is a linearization of the dependence of the revolution period on momentum, a cyclic accelerator with zero linear dependence on momentum but significant higher-order dependence on momentum may be considered to be quasi-isochronous. At isochronous conditions, there can be no phase-focusing and direct space charge forces will lead to particle loss over time. In high-intensity isochronous cyclotrons, losses are minimized by using rapid acceleration.