Electron cloud cyclotron resonances in the presence of a short-bunch-length relativistic beam

Computer simulations using the 2D code POSINST were used to study the formation of the electron cloud in the wiggler section of the positron damping ring of the International Linear Collider. In order to simulate an $x\mathrm{\text{−}}y$ slice of the wiggler (i.e., a slice perpendicular to the beam velocity), each simulation assumed a constant vertical magnetic field. At values of the magnetic field where the cyclotron frequency was an integral multiple of the bunch frequency, and where the field strength was less than approximately 0.6 T, equilibrium average electron densities were up to 3 times the density found at other neighboring field values. Effects of this resonance between the bunch and cyclotron frequency are expected to be non-negligible when the beam bunch length is much less than the product of the electron cyclotron period and the beam velocity, for a beam moving at $v\approx c$. Details of the dynamics of the resonance are described.

Figure 1

Buildup of the average density for $B=0.07$ and $B=0.8\mathrm{T}$ ($n=12$ and 137.8, respectively). The horizontal line shows the value of the “equilibrium average density” for $B=0.07\mathrm{T}$.

Figure 2

Equilibrium average density vs $n$, the ratio of the bunch spacing to the cyclotron period. Simulations were done only at the plus sign markers, which have been connected by lines to guide the eye. (a) Simulations for $0\le n\le 189.5$, or $B\le 1.1\mathrm{T}$; (b) $0\le n\le 70$, showing that peaks in density are at integer values of $n$; (c) expansion of the range $0\le n\le 5.5$ in order to show the minima in the center of these peaks. The depression decreases with increasing $n$.

Figure 3

Density distribution for a resonant (0.07 T) and nonresonant (0.8 T) case. Vacuum chamber is circular and fits exactly within the plot.

Figure 4

Electron orbit in the $x\mathrm{\text{−}}z$ plane (plane perpendicular to $B$) showing the effect of the beam kick on the cyclotron radius and gyrocenter. The electron has $x>0$, therefore the beam kick is in the $-x$ direction. Blue is before the kick; red is after. Arrows show rotation of electron velocity by the beam kick.

Figure 5

Change with time of the cyclotron phase angle, plotted at the arrival time for each bunch, for electrons that remain in the system for long times—resonant and nonresonant case. Dynamics are computed nonrelativistically here for simplicity. On the left is $n=12.0$ (resonant case); on the right is $n=11.5$.

Figure 6

Kinetic energy of all electrons in the tracking calculation vs time for $n=12$, showing oscillation of energy due to detuning from resonance caused by changes in the particle mass. Curves end if the electron hits the wall and leaves the system.

Figure 7

Histogram of final kinetic energies for a resonant ($n=12.0$, blue) and nonresonant ($n=11.5$, red) case for electrons that hit the wall during the 0.3 microseconds simulated. Calculated with single particle tracking code with correct relativistic dynamics. Histogram bins are 10 eV wide.

Figure 8

Cyclotron phase angle for each electron at the time of a bunch arrival vs time for $n=12$. Phase angles are pushed by the resonance to 270 degrees, and then move away from this value as the electron mass increase detunes it from resonance. Curves stop when the electron hits the wall and leaves the system.

Figure 9

Energy spectrum from POSINST simulations of electrons hitting the wall for the time interval from 0 to $0.2\mu \mathrm{s}$ for $n=11.5$ (black) and $n=12$ (red).

Figure 10

Histograms for $n=11.5$ (black) and $n=12$ (red) for electrons hitting the wall in POSINST simulations in the time interval from 2 to $3\mu \mathrm{s}$ (which is after the cloud density has reached equilibrium). Shown are the number of electrons hitting the wall vs $\gamma v_{\perp }$ (left) and the cosine of the impact angle with the wall with respect to the normal (right).

Figure 11

Histograms from POSINST simulations showing the number of electrons which hit the wall in the time interval between 1.2 and $3\mu \mathrm{s}$ with a given $\gamma v_{\perp }$ for $n=12$ (left) and $n=11.5$ (right). Each curve shows the $v_{\perp }$ distribution function for a different range of $x$: $0\le x\le 1\mathrm{mm}$ (red), $1\le x\le 2\mathrm{mm}$ (blue), $2\le x\le 7\mathrm{mm}$ (green), $7\le x\le 10\mathrm{mm}$ (magenta), $10\le x\le 15\mathrm{mm}$ (cyan), $15\le x\le 20\mathrm{mm}$ (black).

Figure 12

Color contour plots of the average energy of electrons, averaged over the time period of the simulation ($0–3\mu \mathrm{s}$) shown as a function of $x$ and $y$ for $n=11.5$ (left) and $n=12.0$ (right) from POSINST simulations.

Figure 13

SEY vs impact energy for our parameters.