Chromaticity effects on head-tail instabilities for broadband impedance using two particle model, Vlasov analysis, and simulations

Effects of the chromaticity on head-tail instabilities for broadband impedances are comprehensively studied, using the two particle model, the Vlasov analysis and computer simulations. We show both in the two particle model and the Vlasov analysis with the trapezoidal (semiconstant) wake model that we can derive universal contour plots for the growth factor as a function of the two dimensionless parameters: the wakefield strength, $\mathrm{\Upsilon }$, and the difference of the betatron phase advances between the head and the tail, $\chi$. They reveal how the chromaticity affects strong head-tail instabilities and excites head-tail instabilities. We also apply the LEP (Large Electron-Positron Collider) broadband resonator model to the Vlasov approach and find that the results are in very good agreement with those of the trapezoidal wake model. The theoretical findings are also reinforced by the simulation results. The trapezoidal wake model turns out to be a very useful tool since it significantly simplifies the time domain analysis and provides well-behaved impedance at the same time.

Figure 1

The complex function $f(\chi )$ as a function of $\chi$. The real and the imaginary parts of $f(\chi )$, namely, $h(\chi )$ and $g(\chi )$, are denoted by the blue solid and the blue dashed lines, respectively.

Figure 2

Three-dimensional contour plot for the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ for positive $\chi$.

Figure 3

Flat contour plot for the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ for positive $\chi$.

Figure 4

Flat contour plot for the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ for negative $\chi$.

Figure 5

The particle motion in the square well potential above the transition.

Figure 6

The complex function $f_s(\chi )$. The real and the imaginary parts of $f_s(\chi )$ are denoted by the green solid and the green dashed lines, respectively. For comparison, the function $f(\chi )$ for the parabolic potential is also plotted. Its real and imaginary parts are denoted by the blue solid and the blue dashed lines, respectively.

Figure 7

Three-dimensional contour plot in the square well model for the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ for positive $\chi$.

Figure 8

Flat contour plot in the square well model for the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ for positive $\chi .$

Figure 9

The growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ when they are less than 0.5.

Figure 10

Three-dimensional contour plot for the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ from an angle different from Fig. 2.

Figure 11

The trapezoidal model for the constant wake approximation.

Figure 12

The transverse impedance of the trapezoidal wake model. The real and the imaginary parts of the impedance are shown by the solid and the dashed lines, respectively.

Figure 13

The growth factor $g\times T_s$ (of the most unstable mode) as a function of $\mathrm{\Upsilon }$ for various TFCTs ($\mathrm{TFCT}=1$ by the red line, $\mathrm{TFCT}=1.8$ by the green lien, $\mathrm{TFCT}=2$ by the black line, and $\mathrm{TFCT}=2.2$ by the blue line, respectively) at zero chromaticity.

Figure 14

The three-dimensional contour plot of the growth factor $g\times T_s$ (of the most unstable mode) as a function of $\mathrm{\Upsilon }$ and $\chi$ at the rms bunch length of 2 cm in the trapezoidal wake model for the Vlasov analysis with the Gaussian bunch distribution.

Figure 15

The flat contour plot of the growth factor $g\times T_s$ (of the most unstable mode) as a function of $\mathrm{\Upsilon }$ and $\chi$ at the rms bunch length of 2 cm in the trapezoidal wake model for the Vlasov analysis with the Gaussian bunch distribution.

Figure 16

The flat contour plot of the growth factor $g\times T_s$ (of the most unstable mode) as a function of $\mathrm{\Upsilon }$ and negative $\chi$ at the rms bunch length of 2 cm in the trapezoidal wake model for the Vlasov analysis with the Gaussian bunch distribution.

Figure 17

The three-dimensional contour plot of the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ for $\mathrm{TFCT}=4$ at the rms bunch length of 2 cm in the trapezoidal wake model for the Vlasov analysis with the Gaussian bunch distribution.

Figure 18

The three-dimensional contour plot of the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ at the rms bunch length of 4 cm in the trapezoidal wake model for the Vlasov analysis with the Gaussian bunch distribution.

Figure 19

The three-dimensional contour plot of the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ at the rms bunch length of 2 cm in the LEP resonator model for the Vlasov analysis with the Gaussian bunch distribution.

Figure 20

The flat contour plot of the growth factor $g\times T_s$ as a function of $\mathrm{\Upsilon }$ and $\chi$ at the rms bunch length of 2 cm in the LEP resonator model for the Vlasov analysis with the Gaussian bunch distribution.

Figure 21

The impedances of the two models. The black line is for the trapezoidal wake model (${\sigma }_z=2\mathrm{cm}$), and the red line is for the LEP broadband resonator impedance model, respectively. The real and the imaginary parts of the impedance are denoted by the solid and the dashed lines, respectively.

Figure 22

The wake potentials of the two models. The black line is for the trapezoidal wake model (${\sigma }_z=2\mathrm{cm}$), and the red line is for the LEP broadband resonator impedance model, respectively.

Figure 23

The growth factor $g\times T_s$ as a function of $\chi$ at $\mathrm{\Upsilon }=2$. The black line is the moscow result, while the green line is the simulation result, for the trapezoidal wake model. The red line is the moscow result for the LEP broadband resonator model at the bunch length of 2 cm and at the bunch current of 0.78 mA that corresponds to $\mathrm{\Upsilon }=2$. The blue line is the simulation result for the LEP broadband resonator model.

Figure 24

The growth factor $g\times T_s$ as a function of $\chi$ at $\mathrm{\Upsilon }=1.5$. The black line is the moscow result, while the green line is the simulation result, for the trapezoidal wake model. The red line is the moscow result for the LEP broadband resonator model at the bunch length of 2 cm and at the bunch current of 0.585 mA that corresponds to $\mathrm{\Upsilon }=1.5$. The blue line is the simulation result for the LEP broadband resonator model.

Figure 25

The growth factor $g\times T_s$ as a function of $\chi$ according to the two particle model. The blue and the green lines are for the $\mathrm{\Upsilon }=2.2$ and $\mathrm{\Upsilon }=1.7$, respectively.