course 4

Space Charge Effect: 

Space Charge Effect Up to now single particle longitudinal dynamics has been considered. In a real beam (or bunch), with many particles, each particle will suffer the repulsive forces from the others since they have the same electrical charge. This intrinsic effect is however important only at low energies and vanishes for ultra-relativistic beams where magnetic forces compensate electric forces. The space charge forces affect the longitudinal dymamics (as well as the transverse one). Since request for higher and higher intensities, at low and medium energies, is driving the next future the space charge phenomenon needs particular attention.

Space Charge Fields in a Bunch: 

Space Charge Fields in a Bunch Assume a uniform particle distribution inside a 3-D ellipsoidal beam shape. In the moving frame of the bunch the S.C. force is purely electrostatic; the field components can be obtained analytically solving Poisson equation. The a’s represent the 3 half axis in the rest frame

Space Charge Fields in a Bunch (2): 

Space Charge Fields in a Bunch (2) The 2-D ellipsoid case (“cigar-like” bunch): The elliptic integral for the longitudinal direction reduces to: Where the a’s are respectively the bunch radius and the half bunch length in the laboratory system. Solving the integral:

Longitudinal Space Charge Force: 

Longitudinal Space Charge Force The force acting on a single particle having a longitudinal position z with respect to the centroid, in the Lab. system, is:

Longitudinal Dynamics with Space Charge: 

Longitudinal Dynamics with Space Charge In the Lab. system, neglecting transverse motion: Space Charge is a defocusing effect leading to bunch lengthening

Longitudinal Space Charge in a Linac: 

Longitudinal Space Charge in a Linac Adding the focusing effect from the RF: At low energy and high current the space charge effect can be dramatic. Increasing the RF power is expensive. Particular attention is to be given to the new generation of high current proton linacs.

Longitudinal Space Charge in Synchrotron: 

Longitudinal Space Charge in Synchrotron Energy deviation of a particle with respect to the reference one: since and Derivation with respect to time leads to:

Longitudinal Space Charge in Synchrotron (2): 

Longitudinal Space Charge in Synchrotron (2) The second order energy equation can then be writen: with: showing that: - below transition (η > 0) the S.C. effect is defocusing - above transition (η < 0) the S.C. effect is focusing The later is often referred to “ negative mass effect”

Longitudinal Space Charge in Synchrotron (3): 

Longitudinal Space Charge in Synchrotron (3) In the case of a “cigar-type” beam, following Reiser’s book, one can introduce a new form factor: Leading to: where N is the number of particles and is the classical electron radius. As can be seen the S.C. factor varies like while the corresponding RF factor varies like . Though the S.C. effect decreases rapidly with energy, special care has to be taken in the vicinity of transition energy (dilution).

Radio-Frequency Gun: 

Radio-Frequency Gun Ez Photo-cathode Specifically designed for high intensity, low energy, electron beam; a multi-cells high Q cavity provides a large electric field that rapidly accelerates the beam to ultra-relativistic energy, hence reducing the space charge effect; it also bunches the beam but giving large energy spread. Generally a short pulse laser hits a photo-cathode to generate short electrons pulses.

Radio-Frequency Quadrupole: 

Radio-Frequency Quadrupole Specifically designed for intense low velocity protons (or ions) beams; it both accelerates and focus to control space charge effects (see A. Lombardi lecture) 4 vanes resonator that provides a quadrupolar symmetry which gives a transverse E gradient for focusing. Modulated pole shapes provide a longitudinal E field for acceleration and bunching.

Acceleration of Intense Beams: 

Acceleration of Intense Beams Obviously the accelerated beam gets its energy from the stored energy in the cavity: PRF = Pdiss. + Pbeam The cavity voltage is the vector sum of the voltage due to the generator and the “beam loading”: Vt = VRF + Vbeam = ZRF Ig + Zb Ib Under proper matching and tuning (cavity on-resonance) the impedance is just the shunt impedance R. Since the beam loading is just like a power loss one can introduce a corresponding Q factor, Qb . The loaded Q becomes:

Acceleration of Intense Beams (2): 

Acceleration of Intense Beams (2) Equivalent circuit with beam Ib Vt During acceleration a synchronous phase is established between the current and the voltage: The resulting effect is a detuning of the cavity ; a feed back system is used to compensate for that. Optimum power transfer to the cavity and beam is made by proper matching of the power supply to the cavity through a feeder and a coupling loop.