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G. Parzen

Publications and source records attributed to G. Parzen.

12 recordsLinked to original sources
arXiv2003

Island Resonances

Resonances driven by the error field multipoles in the magnets often show themselves as islands in phase plots of the particle motion. In order to be able to measure the strength of the island resonance, and then to correct it, it is helpful to study how these resonances show themselves. In particular, one can study how the presence of the island resonances distort the tune dependence on the amplitude, and the emittance growth caused by the island resonance. Islands in 4-dimensional phase space, unlike islands in 2-dimensional phase space, are not easy to visualize. This study will show how to visualize the islands in 4-dimensional phase space by studying the tune dependence on the amplitude of the particle motion. These effects are to some extent measureable, and can lead to a way to correct the resonance.

physics.acc-ph↗
arXiv2000

Prediction of long term stability by extrapolation

This paper studies the possibility of using the survival function to predict long term stability by extrapolation. The survival function is a function of the initial coordinates and is the number of turns a particle will survive for a given set of initial coordinates. To determine the difficulties in extrapolating the survival function, tracking studies were done to compute the survival function. The survival function was found to have two properties that may cause difficulties in extrapolating the survival function. One is the existence of rapid oscillations, and the second is the existence of plateaus. It was found that it appears possible to extrapolate the survival function to estimate long term stability by taking the two difficulties into account. A model is proposed which pictures the survival function to be a series of plateaus with rapid oscillations superimposed on the plateaus. The tracking studies give results for the widths of these plateaus and for the seperation between adjacent plateaus which can be used to extrapolate and estimate the location of plateaus that indicate survival for longer times than can be found by tracking.

physics.acc-ph↗
arXiv1999

Theory of the tune shift due to linear coupling

The presence of skew quadrupole fields will linearly couple the x and y motions. The x and y motions can then be written as the sum of two normal modes >. This paper presents analytical perturbation theory results for the tunes of the normal modes. The results for the normal mode tunes are first found correct to lowest order in the skew quadrupole fields. The results are then carried one step further to include the next higher order terms in the skew quadrupole fields. These analytical results show that for the higher order shift in the tune, the important harmonics of the skew quadrupole field are the harmonics near the sum of the tunes. However the harmonics closest to the sum of the tunes do not contribute to the higher order tune splitting, the seperation of the tunes, as they shift the two tunes about equally.This results in a lack of a dominant harmonic for the higher order part of the tune splitting, which complicates the understanding and correction of the higher order part of the tune splitting.

physics.acc-ph↗
arXiv1998

Symplectic tracking using point magnets and a reference orbit made of circular arcs and straight lines

Symplectic tracking of beam particles using point magnets is achieved using a reference orbit made of circular arcs and straight lines that join smoothly with each other. For this choice of the reference orbit, results are given for the transfer functions, transfer matrices, and the transit times of the magnets and drift spaces. These results provide a symplectic integrator, and allow the linear orbit parameters to be computed by multiplying transfer matrices. It is shown that this integrator is a second-order integrator, and that the transfer functions can be derived from a hamiltonian.

physics.acc-ph↗
arXiv1998

The Dynamic Aperture and the High Multipole Limit

Tracking studies have indicated that for a lattice whose elements all have a single field multipole present, all having the same order k, the dynamic aperture approaches a non zero limit when k becomes very large. The dynamic aperture and other properties of the lattice, as k becomes large, will be called the high multipole limit. It will be shown that the high multipole limit provides a reasonable estimate of the dynamic aperture of an accelerator, and the other properties of the high multipole limit found below are useful for understanding the stability of the accelerator. The high multipole limit is easily computed and it also provides an estimate of how much can be gained by correcting the lower field multipoles. The above results will be illustrated by tracking studies done with a simple one cell lattice, and with a RHIC lattice having six low beta insertions.

physics.acc-ph↗
arXiv1996

Orbit Dynamics for Unstable Linear Motion

A treatment is given of the orbit dynamics for linear unstable motion that allows for the zeros in the beta function and makes no assumptions about the realness of the betatron and phase functions. The phase shift per turn is shown to be related to the beta function and the number of zeros the beta function goes through per turn. The solutions of the equations of motion are found in terms of the beta function.

acc-phys↗
arXiv1996

Particle Motion in the Stable Region Near an Edge of a Linear Sum Resonance Stopband

This paper studies the particle motion when the tune is in the stable region close to the edge of linear sum resonance stopband. Results are found for the tune and the beta functions. Results are also found for the two solutions of the equations of motion. The results found are shown to be also valid for small accelerators where the large accelerator approximation may not be used.

acc-phys↗
arXiv1995

Particle Motion in the Stable Region Near an Edge of a Linear Half-Integer Stopband

This paper studies the motion of a particle whose tune is inside and near a linear half-integer stopband. Results are found for the tune and beta functions in the stable region close to an edge of the stopband. It is shown that the eigenvalues and the eigenfunctions of the transfer matrix are real inside the stopband. All the results found are also valid for small accelerators where the large accelerator approximation is not used.

acc-phys↗
arXiv1995

Linear Parameters and the Decoupling Matrix for Linearly Coupled Motion in 6 Dimensional Phase Space

It will be shown that starting from a coordinate system where the 6 phase space coordinates are linearly coupled, one can go to a new coordinate system where the motion is uncoupled by means of a linear transformation. The original coupled coordinates and the new uncoupled coordinates are related by a 6x6 matrix, R. R will be called the decoupling matrix. It will be shown that of the 36 elements of the 6x6 decoupling matrix R, only 12 elements are independent. This may be contrasted with the results for motion in 4-dimensional phase space, where R has 4 independent elements. A set of equations is given from which the 12 elements of R can be computed from the one period transfer matrix.This set of equations also allows the linear parameters for the uncoupled coordinates to be computed from the one period transfer matrix. An alternative procedure for computing the linear parameters and the 12 independent elements of the decoupling matrix R is also given which depends on computing the eigenvectors of the one period transfer matrix. These results can be used in a tracking program, where the one period transfer matrix can be computed by multiplying the transfer matrices of all the elements in a period, to compute the linear parameters and the elements of the decoupling matrix R.

acc-phys↗
arXiv1995

Linear Orbit Parameters for the Exact Equations of Motion

This paper defines the beta function and other linear orbit parameters using the exact equations of motion. The orbit functions are redefined using the exact equations. Expressions are found for the transfer matrix and the emittances. Differential equations are found for the beta function and the eta function. New relationships between the linear orbit parameters are found.

acc-phys↗
arXiv1995

Normal Mode Tunes for Linear Coupled Motion in Six Dimensional Phase Space

The motion of a particle in 6-dimensional phase space in the presence of linear coupling can be written as the sum of 3 normal modes. A cubic equation is found for the tune of the normal modes, which allows the normal mode tunes to be computed from the 6x6 one turn transfer matrix. This result is similar to the quadratic equation found for the normal mode tunes for the motion of a particle in 4-dimensional phase space. These results are useful in tracking programs where the one turn transfer matrix can be computed by multiplying the transfer matrices of each element of the lattice. The tunes of the 3 normal modes, for motion in 6-dimensional phase space, can then be found by solving the cubic equation. Explicit solutions of the cubic equation for the tune are given in terms of the elements of the 6x6 one turn transfer matrix.

acc-phys↗