High-intensity effects on longitudinal bunch merging in hadron synchrotrons

Longitudinal rf manipulation schemes have been widely employed for achieving various beam experiments and applications in heavy ion or proton (hadron) synchrotrons. For high-intensity hadron beams, longitudinal space charge and cavity beam loading play a key role in beam intensity limitations since they may cause beam oscillations and longitudinal emittance growth. Efficient schemes to compress such intense bunched beams and minimize the emittance blow-up during those manipulations are of practical concern. In this article, the behavior of the particle distribution in the presence of space charge and beam loading during bunch merging is investigated via a generalized elliptical bunch model and particle-in-cell (PIC) simulations. Possible schemes to minimize these intensity effects are discussed. As an application, parameters for the longitudinal rf manipulation scenario in the upcoming second phase of the China Spallation Neutron Sources (CSNS-II) are proposed. It is shown that for (slow cycling) storage rings, with an optimized set of longitudinal parameters, the emittance growth due to intensity effects can be largely dampened and a high compression efficiency is achieved. For rapid cycling synchrotrons, fast bunch merging can be achieved via a desynchronization method. The agreement between the analytical elliptical model and the PIC simulation results indicates that the extended model can be employed for the study of the intensity effects during longitudinal rf manipulations in hadron synchrotrons.

Figure 1

Proposed scheme of the longitudinal proton beam manipulations in one beam cycle for the upcoming CSNS-II. The injected beams are bunched into two buckets with the initial harmonic number $h_i=2$. Afterward, the two bunches are accelerated via increasing rf voltage. Finally, the two bunches are merged into one single bunch before extraction under the dual rf system with the initial harmonic $h_i=2$ and the final harmonic $h_f=1$.

Figure 2

Schematic of the initial positions of the two bunches (red shade) and the buckets in the initial harmonics $h_{\mathrm{i}}$ (solid blue) and in the final harmonic $h_{\mathrm{f}}$ (dashed blue). The red solid and dashed curves indicate the rf potential in the initial harmonic and in the final harmonic, respectively.

Figure 3

Comparison of bunch merging with a constant bunching factor (upper) and with a constant bunch ratio (lower).

Figure 4

Schematic of the matched rf voltage amplitude ramps for initial harmonic (blue) and final harmonic (red) with (dashed line) and without (solid line) space charge during bunch merging. ($t_m$ represents the merging time.).

Figure 5

The beam-loading voltages (solid blue) induced by one bunch (upper plot) and two bunches (lower plot) in the dual harmonic rf system of CSNS-II/RCS, compared with the space-charge voltages (green). The dotted and dashed blue lines represent the beam-loading voltages (wakefields) from $h_f=1$ and $h_i=2$ harmonic systems, respectively. The red lines denote the beam-loading voltages acting on the bunches.

Figure 6

The bucket trajectories (dashed line) and bunch boundaries (solid line) without space charge and beam loading (in black); matched to space charge (in blue); matched to beam loading (in green); and matched to both space charge and beam loading (in red), calculated via using iteration scheme from the generalized elliptical bunch model. The two points in black and in red denote the centers of the two buckets without and with beam loading, respectively.

Figure 7

Evolution of bunch center position mismatched (in red) and matched (in blue) to beam loading: The red curve indicates the mismatched dipole oscillation because of beam loading, while the blue curve represents the matched case with a shift $\mathrm{\Delta }{z}_{\mathrm{bl}}$ acting on bunch center. For comparison, the ideal case in the absence of beam loading is also shown (in black).

Figure 8

Schematic drawing of the shifts $\mathrm{\Delta }{z}_{\mathrm{bl},1}$, $\mathrm{\Delta }{z}_{\mathrm{bl},2}$ on the two bunches for beam-loading matching and the shift $\mathrm{\Delta }{z}_{\mathrm{h}}$ on the harmonics for symmetry compensation: The red shade areas represent the bunches without beam-loading-matching shift; the red dashed ellipses indicate the shifted bunches with beam-loading matching; the solid blue line denotes the bucket on the final harmonic $h_f$ with its center $o$; the dashed blue ellipse is the shifted bucket with its center $o^{\prime }$ to keep the symmetry of the two bunches during bunch merging.

Figure 9

Snapshots of phase-space distribution during bunch merging in the absence of intensity effects with four different merging times $t_m=20T_{\mathrm{s}0}$, $5T_{\mathrm{s}0}$, $3T_{\mathrm{s}0}$, and $1T_{\mathrm{s}0}$. The three panels in each row from left to right represent three time instants: at the beginning ($t=0$), in the middle ($t=t_{\mathrm{m}}/2$), and at the end ($t=t_{\mathrm{m}}$) of the merging process. The blue lines denote the separatrix orbit in the dual harmonic system. The green and the yellow lines represent the separatrix orbit in the harmonic $h_{\mathrm{i}}=2$ and $h_{\mathrm{f}}=1$, respectively, and the light blue (cyan) lines denote the trajectories of the constant Hamiltonian at the bunch boundary.

Figure 10

Comparison of the final normalized emittance with different merging times, $20T_{\mathrm{s}0}$, $10T_{\mathrm{s}0}$, $5T_{\mathrm{s}0}$, $3T_{\mathrm{s}0}$, and $1T_{\mathrm{s}0}$, in three cases: in the absence of space charge (solid blue), in the presence of space charge but without space-charge matching (red dashed); in the presence of space charge and with space-charge matching (red solid).

Figure 11

The normalized emittance of two bunches after bunch merging in the presence of beam loading and space charge with different merging times $t_m=2T_{\mathrm{s}},5T_{\mathrm{s}},10T_{\mathrm{s}},15T_{\mathrm{s}},20T_{\mathrm{s}},25T_{\mathrm{s}}$ and $30T_{\mathrm{s}}$: (i) without shifts (dashed red); (ii) with matching shift, without symmetry shifts (solid red); (iii) with both shifts (solid blue).

Figure 12

Snapshots of phase-space distributions at the initial ($t=0$), half ($t=t_{\mathrm{m}}/2$), three quarter ($t=3t_{\mathrm{m}}/4$) and the end ($t=t_{\mathrm{m}}$) of bunch merging with merging time of $10T_{\mathrm{s}0,\mathrm{i}}$, in the presence of both space charge and beam loading for three cases. Upper panels (1) to (4): simulations without matching shift and without symmetry shift; Middle panels (5) to (8): with matching and without symmetry shift; Lower panels (9) to (12): with both matching and symmetry shift. The purple lines are the separatrix orbit of the combined harmonics affected by the beam loading and space charge, compared with the ideal separatrix without beam loading and space charge (in blue). The two curves in cyan and purple denote two trajectories of constant Hamiltonian including space charge and beam loading. For comparison, panel(9) to (12) displays the fast bunch merging with $t_{\mathrm{m}}=2T_{\mathrm{s}0,\mathrm{i}}$.

Figure 13

The magnetic field ramping in one cycling period in CSNS-II/RCS. The dotted lines denote the amplitude of the maximum $\widehat{B}$ and the ac component $B_{\mathrm{ac}}$, respectively ($B_{\mathrm{ac}}$ is not confused with the minimum magnetic field $\widehat{B}-2B_{\mathrm{ac}}$). The dashed red lines represent the duration time of the merging process.

Figure 14

Proposed proton bunch merging process in CSNS/RCS. The solid lines denote the line density profile.

Figure 15

Variation of rms beam length and normalized emittance during bunch merging in CSNS-II/RCS.