Longitudinal beam dynamics studies with space charge in small isochronous ring

Studies of the longitudinal beam dynamics in the small isochronous ring (SIR) at Michigan State University revealed a fast, space-charge driven instability that did not fit the model of the negative mass instability. The observed beam behavior can be explained by the transverse horizontal component of the coherent space-charge force and its effect on the longitudinal motion. This force effectively modifies the slip factor, shifting the isochronous point and enhancing the negative mass instability. This paper presents results of numerical and experimental studies of the longitudinal beam dynamics in SIR and proposes a simple analytical model explaining these results.

Figure 1

Small isochronous ring.

Figure 2

Longitudinal bunch profiles measured by the fast Faraday cup after injection and after turns 10 and 20. The profiles measured after 10 and 20 turns are shifted vertically by 0.3 and 0.6, respectively. In these measurements, the beam energy was 20.9 MeV and the peak current was $9.3\mu \mathrm{A}$.

Figure 3

Longitudinal impedance (arbitrary units) in long and short wavelength approximations as a function of the wavelength normalized to the beam diameter, $2a$. The impedance reaches its maximum at $\lambda \approx 5a$. The $b/a$ ratio is equal to 5.

Figure 4

(a) Schematic drawing of a horizontal beam shape distortion caused by energy modulation. Also, the sketch depicts the $x$ component of the coherent field and one of the beam slices that are used to calculate this field. (b) A schematic drawing of a cylindrical beam slice with its longitudinal and transverse fields.

Figure 5

Factor $1-ka·K_1(ka)$ as a function of the wavelength normalized to the beam diameter, $2a$.

Figure 6

Results of simulations of the beam dynamics in SIR for three different peak intensities: 5, 10, and $20\mu \mathrm{A}$. The bare machine optics is isochronous. Each frame contains a projection of the bunch charge density on the median plane for a given turn and intensity. The vector of velocity is directed from the top of the figure to the bottom. The horizontal axis of each frame corresponds to $x$ measured in centimeters from the center of the ring. The vertical axis is along the beam trajectory. The size of each frame is $5(x)\times 45(z){\mathrm{cm}}^2$.

Figure 7

Linear particle density profiles for turns 7, 11, and 15. The average level equal to 430 was subtracted. Only the central $\pm 15\mathrm{cm}$ part of a 40 cm long bunch is shown. The charge density is in arbitrary units.

Figure 8

Spectra of linear charge density profiles shown in Fig. 7.

Figure 9

Logarithm of the amplitude of harmonics with $\lambda =0.6$, 0.75, 1, 1.5, and 3 cm as a function of turn number. The harmonic with $\lambda =1.5\mathrm{cm}$ exhibits the highest growth rate.

Figure 10

Amplitude growth rate of harmonics $\lambda =0.6$, 0.75, 1, 1.5, and 3 cm. The growth rate shown for four different peak intensities: 5, 10, 15, and $20\mu \mathrm{A}$.

Figure 11

Amplitude growth rate for harmonics with the wavelength $\lambda =0.6$, 0.75, 1, 1.5, and 3 cm normalized to the peak beam current. The normalized growth rate shown for four different peak intensities: 5, 10, 15, and $20\mu \mathrm{A}$.

Figure 12

Growth rate of the harmonic with $\lambda =1.5\mathrm{cm}$ for the isochronous optics and the optics with the negative slip factor $-0.04$. The simulated data (solid lines with points) is fitted with corresponding fits: $y=0.028x$ and $y=0.028x\sqrt{1-7/x}$. Additionally, growth rates calculated directly from Eqs. (14, 16) for $a=0.5\mathrm{cm}$ and $\lambda =1.5$ are shown. The growth rates predicted by the model exceed the simulated growth rates by approximately 20%.

Figure 13

Simulated (in grey) and experimentally measured (in red) number of linear charge density peaks vs turn number for three different beam intensities: 5, 10, and $20\mu \mathrm{A}$. Experimental and simulated data closely overlap.

Figure 14

(a) The number of peaks of the linear charge density as a function of turn number for three different peak intensities: 5, 10, and $20\mu \mathrm{A}$. (b) The same data plotted as a function of a product of turn number and beam intensity (equivalent turn number).

Figure 15

(a) The number of peaks of the linear charge density for three different initial bunch lengths: 20, 40, and 80 cm. (b) The same data normalized to the initial bunch length measured in nanoseconds. It is obvious that the growth rate of the instability does not depend on the bunch length.