Beam halo in high-intensity hadron accelerators caused by statistical gradient errors

The particle-core model for a continuous cylindrical beam is used to describe the motion of single particles oscillating in a uniform linear focusing channel. Using a random variation of the focusing forces, the model is deployed as proof of principle for the occurrence of large single particle radii without the presence of initial mismatch of the beam core. Multiparticle simulations of a periodic 3D transport channel are then used to qualify and quantify the effects in a realistic accelerator lattice.

Figure 1

Particle and core envelopes for initial mismatch, $\mu =0.6$, $x(0)=0.9r_0$, and $\tau =0.7$.

Figure 2

Stroboscopic plot for $\mu =0.95$, $0.2\le x(0)/r_0\le 3.5$, and $\tau =0.8$.

Figure 3

Stroboscopic plot for $\mu =0.6$, $0.2\le x(0)/r_0\le 3.5$, and $\tau =0.8$.

Figure 4

Particle and core envelopes for a matched beam with 1% rms statistical focusing gradient errors $x(0)/r_0=1.2$, $\tau =0.8$.

Figure 5

Maximum envelope values for core and single particle oscillations in the case of a matched beam with 1% rms statistical focusing gradient errors $0.2\le x(0)/r_0\le 3.5$, $\tau =0.8$.

Figure 6

Maximum amplitude growth for single particles over 100 zero-current betatron periods as a function of their initial radius and tune depression $\tau$. Each dot represents the averaged results of 1000 runs with different error sets for 1% rms error.

Figure 7

Maximum amplitude growth for single particles over 100 zero-current betatron periods as a function of their initial radius and of the rms error amplitude. Each dot represents the averaged results of 1000 runs with different error sets for a tune depression of $\tau =0.8$.

Figure 8

Maximum amplitude growth for single particles as a function of their initial radius and the length of the transport channel (in zero-current betatron periods). Each dot represents the averaged results of 1000 runs with different error sets for a tune depression of $\tau =0.8$ and an rms error amplitude of 1%.

Figure 9

Wave numbers for core and single particle oscillations. Single particle wave numbers are multiplied by 2 and depict particles starting with different initial radii (1.1–2.7 times the matched core radius), $\tau =0.8$, rms error amplitude: 1%.

Figure 10

Stroboscopic plot for a single particle with $x(0)/r_0=1.2$, 1% rms error, and 1000 zero-current betatron periods.

Figure 11

Probability for single particles to reach final radii with large amplitudes; 200 and 800 zero-current betatron periods. The black bars represent one out of a total of 1000 simulations.

Figure 12

Averaged fraction of particles exceeding $n{\epsilon }_{\mathrm{rms},t}$ for transport channels of different length (100 focusing periods correspond to $\approx 10.7$ zero-current betatron periods), 6D Gauss with 1% (rms) quadrupole gradient errors, 500 simulations, normalized to transverse input emittance.

Figure 13

Averaged fraction of particles exceeding $n{\epsilon }_{\mathrm{rms},t}$ for statistical error runs normalized to the final rms emittance of each case. 500 simulations, 6D Gauss, 200 focusing periods, and $\Delta \epsilon \to \mathrm{rms}$ emittance growth.

Figure 14

Mismatched transverse core radii over matched core radii for 6D waterbag and Gauss, 200 focusing periods, and fast mode excitation.

Figure 15

Averaged fraction of particles exceeding $n{\epsilon }_{\mathrm{rms},t}$ for statistical error runs normalized to the final rms emittance of each case. 500 simulations, 6D Gauss/waterbag, 200 focusing periods, and $\Delta \epsilon \to \mathrm{rms}$ emittance growth.