Ultrafast electron diffraction (UED) relies on short, intense pulses of electrons, which because of Coulombic repulsion will expand and change shape as they propagate. While such pulse expansion has been studied in other contexts, efforts to model this effect for typical UED parameters have only arisen fairly recently. These efforts have yielded accurate predictions with very simple models, but have left a number of unexplained results (such as the development of a linear self-similar profile with sharply defined end points). The present work develops a series of models that gradually incorporate more physical principles, allowing a clear determination of which processes control which aspects of the pulse propagation. This will include a complete analytical solution of the one-dimensional problem (including a fundamental limitation on temporal resolution), followed by the gradual inclusion of two-dimensional and inhomogeneous effects. Even very simple models tend to capture the relevant on-axis behavior t...

Femtosecond electron pulse propagation for ultrafast electron diffraction

The addition of a higher harmonic RF system to the main system allows a control of the synchrotron frequency, the spread in synchrotron frequency and the bunch length. Adjustment of the higher harmonic system so as to reduce the slope of the RF wave to zero at the bunch centre leads to a longer bunch and a greatly increased spread in synchrotron frequency. This increases the Landau damping against: longitudinal coupled bunch instabilities. The motion of single particles in this highly non-linear potential is calculated numerically as well as analytically (by making some approximations). The dependence of the synchrotron frequency on amplitude and the forms of the synchrotron oscillations and the RF bucket are calculated. Finally the bunch shape and the distribution of particles in Qs are calculated for electron bunches.

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Beam Dynamics in a Double RF System

Thermal wave model for nonlinear longitudinal dynamics in particle accelerators

Beam instabilities cover a wide range of effects in particle accelerators and they have been the subjects of intense research for several decades. As the machines performance was pushed new mechanisms were revealed and nowadays the challenge consists in studying the interplays between all these intricate phenomena, as it is very often not possible to treat the different effects separately. The aim of this paper is to review the main mechanisms, discussing in particular the recent developments of beam instability theories and simulations.

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Beam Instabilities in Hadron Synchrotrons

This distribution fits well with distributions observed in proton synchrotrons and makes several analytical calculations for bunched beams in longitudinal phase space possible. For any shape of the focusing force the line density becomes proportional to the potential well. Self-forces caused by space-charge and inductive wall impedances are thus proportional to the external force, making calculation of bucket area reduction and bunch lengthening easy. The microwave instability threshold, as given by the Keil-Schnell criterion with local values for current and energy spread, is independent of the azimuthal position along the bunch, and again analytical formulae are possible even for strongly non-linear focusing forces. The relative magnitude of the self-force and the microwave threshold turn out to be closely related, as the self-force is always 40% of the external force when the microwave threshold is reached. The classical longitudinal space-charge limit can therefore only be reached within a factor of 0.4. Other calculations with this "natural" distribution include analytical formulae for the rigid dipole mode threshold, and creation of flat-topped bunches with reduced peak line density resulting in a higher transverse space-charge limit.

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Bunches with Local Elliptic Energy Distributions

Space charge and reactive wall impedance create longitudinal forces inside the bunch which change the incoherent phase oscillation frequency, the bunch length and the size of the RF-bucket. These effects have been investigated with bunched beams in the ISR. By measuring the shift of the quadrupole mode phase oscillation frequency, the strength of the self-forces was determined. The inductive wall is dominant and its impedance (divided by the mode number) was measured to be |Z|/n z 26 Ohms. An increase of bunch length with current was measured. It can be explained by the inductive impedance up to a certain current; beyond that an excessive, unexplained bunch lengthening occurs. The reduction of the bucket size affects the stacking process. By correcting for it, an increased density of the stacked beam was achieved. Ez = -e az [4 g0 dz (1)

/pdf/effects-of-space-charge-and-reactive-wall-impedance-on-402ntmeas9.pdf

Effects of Space Charge and Reactive Wall Impedance on Bunched Beams

Microwave instabilities occur in bunched beams in the ISR leading to a dilution of the phase space density and limiting the longitudinal density of the stacked beams. According to D. Boussard this instability can be described as a coasting beam instability inside bunches. Experimental investigations of this microwave instability support this theory and give a high frequency impedance ?ZL?/n ? 14 ohms. Injecting large currents in bunches of large area increases the threshold of this instability. The larger currents can produce coupled bunch mode instabilities which can be cured by a higher harmonic cavity.

/pdf/longitudinal-instabilities-of-bunched-beams-in-the-isr-3w16vifngo.pdf

Longitudinal Instabilities of Bunched Beams in the ISR

Investigations and cures of longitudinal instabilities of bunched beams in the ISR

A brief description of the low power and controls electronics for running the 128 cavity assemblies in the initial phase of LEP is given. Virtually identical hardware will be used for the future extension of LEP with superconducting niobium cavities. One complete RF unit consisting of 16 cavities and all associated drive and controls electronics has been successfully tested.1

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The LEP Radio-Frequency Low Power System

The effect of phase noise on a tightly bunched proton beam is investigated taking into account the frequency spread in the beam, the wall impedance and the phase feedback loop. Under normal conditions the measured dilution rates in the ISR correspond typically to a doubling of the bunch length in one to a few hours, and are consistent with the measured noise spectra. Amplitude noise is not investigated.

/pdf/longitudinal-bunch-dilution-due-to-rf-noise-1urpezcipa.pdf

S. Hansen

Papers

Effects of Space Charge and Reactive Wall Impedance on Bunched Beams

Longitudinal Instabilities of Bunched Beams in the ISR

Investigations and cures of longitudinal instabilities of bunched beams in the ISR

The LEP Radio-Frequency Low Power System

Longitudinal Bunch Dilution Due to RF Noise