Longitudinal stability in multiharmonic standing wave linacs

Accelerating cavities that excite multiple modes at integer harmonics of the fundamental frequency have the potential to be used to suppress the onset of rf breakdown and reduce the pulsed surface heating at high accelerating gradients. Understanding the effect of an additional harmonic cavity mode on the longitudinal beam dynamics is important to their development and use. A Hamiltonian that describes the longitudinal motion of a particle as it traverses a chain of multiharmonic cavities has been derived and is applied to the case of a second harmonic cavity. The Hamiltonian is based upon formalisms found in literature for the fundamental harmonic and is extended to include different longitudinal field distributions and harmonic frequencies. The study initially explores the longitudinal motion for moderate accelerating gradients with high-$\beta$ protons, as this will allow fundamental properties of the stable region (acceptance and shape of the rf bucket) to be determined. High accelerating gradients are also investigated but the focus will be on phase stability throughout. This work concludes by considering the longitudinal dynamics of a modified European Spallation Source accelerator, comprised of multiharmonic cavities that has specifications broadly consistent with the accelerator.

Figure 1

Top: Energy gain as a function of phase offset for a particle traversing a single cavity. Middle: Potential for separatrix. Bottom: Phase space contours in black, separatrix in red and stable orbits in blue.

Figure 2

Comparison of the Hamiltonian found in Eq. (14) with results from a particle tracker based on the fundamental principle of the energy gain of a particle as it traverses a cavity, found in Eq. (7). The peak amplitude is in agreement to within 0.5%.

Figure 3

A single input acceptance is shown for $E_0=10\mathrm{MV}/\mathrm{m}$. The red curve is the boundary of the input acceptance and the blue and black curves are particle trajectories with different initial positions. The trajectory can move beyond the boundary as the boundary only applies to the initial position of the particles. The spiral nature is caused by the adiabatic phase damping due to the increasing ${\beta }_s{\gamma }_s$.

Figure 4

Phase space plots for a cavity exciting both the ${\mathrm{TM}}_{010}$ and ${\mathrm{TM}}_{011}$ modes. A phase shift is gradually applied to the ${\mathrm{TM}}_{011}$ mode and the gradient is kept constant for each step. For all steps, $E_0=10\mathrm{kV}/\mathrm{m}$. The upper plot in each step is the energy gain over the cavity as a function of $\varphi$ and the bottom is the phase space plots, where the red line marks the separatrix.

Figure 5

The normalized acceptance of the rf bucket vs the phase shift of the ${\mathrm{TM}}_{011}$ mode.

Figure 6

Comparison between particle tracking based on fundamental principles with the Hamiltonian from Eq. (27) for ${\varphi }_{nh}=0.5\pi$. The larger contour is for the separatrix, with the smaller trace showing a small phase deviation.

Figure 7

Input acceptance for a ${\mathrm{TM}}_{011}$ second harmonic cavity for $E_0=10\mathrm{MV}/\mathrm{m}$ and an initial energy $W_i=7m$, where $m$ is the proton rest mass. In each case, ${\varphi }_2$ corresponds to respective ${\varphi }_2$ found from the case where no acceleration is present.

Figure 8

Input acceptance for an equivalent ESS linac for a fundamental mode alone (blue) and for a fundamental mode in addition to a second harmonic mode (red). The dashed line represents the input bunch into the equivalent from the end of the RFQ section of the ESS. The initial phases of the input bunches have been shifted to reflect the synchronous phase of each setup.

Figure 9

Phase distribution for the output bunch from the equivalent linac for single mode and multimode cases. The phases have been shifted by their synchronous phases such that they are centered on 0.

Figure 10

Energy distribution for the output bunch from the equivalent linac for single mode and multimode cases.