HEADTAIL program
simulates the interaction between a single proton or positron bunch
and the uniform electron cloud which has built up inside the beam pipe.
The process of build up can be simulated beforehand by means of the
ECLOUD code,
so that the information about the average electron density in the beam
pipe, once the cloud has reached its stationary state, can be herefrom
extracted as a result and subsequently used as an input of the
HEADTAIL program.
The principle of the simulation is illustrated in the following figure.
When studying the single-bunch effects of the electron cloud, only perturbations of the cloud induced by the passing bunch are considered. All the relevant bunch and lattice parameters, as well as the average equilibrium density of the electron cloud along the ring, are basic input parameters for the simulation of the coupled motion between bunch and cloud electrons. For simplicity, the kick approximation is used for the action of the electron cloud on the bunch: the cloud is assumed to be localised at one, or more, definite positions along the ring, s=n × sel with n=0,1,..,(Nint-1). Both the cloud and the bunch are modelled as ensembles of macro-particles (with Np bunch macro-particles and Ne macro-electrons in the cloud). The bunch is also divided into Nsl slices, which interact with the electron cloud after one another and cause the distortion of the initially uniform cloud distribution that can significantly affect the part of the bunch coming later. The interaction between bunch particles and cloud electrons is expressed by the equations of motion:
where the positions of electrons and bunch particles are represented by the vectors xe=(xe, ye) and xp (s)=(xp, yp,z p), z=s-ct being a co-moving longitudinal coordinate; K(s) is the transfer matrix with the focusing strengths between two interaction points; Neme and Npk represent the number of electrons in one macro-electron and the number of particles in the k-th bunch slice, respectively; x offk and
k are the transverse offset
and the rms-size of the k-th slice;
me is the rms-size of a macro-electron,
which is assumed to have finite size; F is expressed by
the Bassetti-Erskine (CERN-ISR-TH/80-06 1980) formula.
The interaction between bunch and electron cloud is simulated following the steps that are shown in the figure above. The macro-electrons have initially a uniform distribution which extends transversely over a region 10 to 20 times larger than the bunch rms-sizes. The initial velocities of the macro-electrons are set to zero. The particles in the bunch are set to have initial Gaussian distributions in each coordinate of their 6-dimensional phase space, and the bunch is subsequently sliced. The bunch slicing is actually repeated each time the interaction starts, because when synchrotron motion is taken into account, particles mix longitudinally. As the bunch slice k interacts with the electron cloud concentrated at the kick point, the particles therein contained and the electrons receive a mutual kick; the perturbed electron cloud acts then on the particles in the next bunch slice k+1, whereas the slice k will be newly kicked by the electron cloud at the next interaction point (after all the bunch particles have been propagated using a linear transport matrix, which can optionally account for the synchrotron motion, and have received a chromaticity kick). Without synchrotron motion, the mechanism does not affect the head of the bunch, which always feels a zero total force from the electrons, but it can drive its tail unstable. The electron cloud configuration is uniformly re-generated at the beginning of each interaction.
In this version of the code, 105 macro-electrons and 3x105 bunch macro-particles are used. The bunch is divided into 50 slices. These parameters can be obviously changed at any time in the
#define lines of the source code.
The macro-electrons can have optionally finite size (and then in the input
file at the line Flag_for_the_e-field_calc_method_(0->no_1->soft_Gauss_2->PIC):
1 has to be set) or their electric field can be calculated by using a particle-in-cell
built-in module (set 2 in the above specified line of the input code).
The interaction starts from the slice containing particles with the largest positive values of z. The model that we have used allows the resolution of the head-tail motion of the bunch, as well as the horizontal and vertical emittance and rms-sizes evolutions, both locally and averaged over the full bunch. Similar simulations were reported before (K. Ohmi and F. Zimmermann, CERN-SL-Note-2000-015 AP 2000) , where the beam was represented by an ensemble of micro-bunches with constant transverse size.