Beam-core oscillations

Let us begin with a coasting beam propagating through a uniformly focusing structure. Such a structure can be used to describe the average dynamic behavior of beams in an alternating gradient focusing channel [31] (i.e., the smooth approximation method). For simplicity, henceforth, we neglect any impedance effects caused by the beam pipe and all the chromatic terms. We adopt x and y to represent the transverse degrees of freedom in the horizontal and vertical directions, respectively, and s the longitudinal coordinate. The “pseudo” Hamiltonian of the beam envelope oscillation in such a transport system is $\begin{array}{ll}{H}_{\text{env}}\hfill & =\frac{1}{2}\left({\sigma }_{\text{p}x}^2+{\sigma }_{\text{p}y}^2+{\kappa }_{x\mathrm{,0}}^2{\sigma }_x^2+{\kappa }_{y\mathrm{,0}}^2{\sigma }_y^2\right)\hfill \\ \hfill & -\frac{1}{2}{K}_{\text{sc}}\ln \left({\sigma }_x+{\sigma }_y\right)+\frac{{\epsilon }_x^2}{2{\sigma }_x^2}+\frac{{\epsilon }_y^2}{2{\sigma }_y^2}\mathrm{,}\hfill \end{array}$ (1) where σ_x,y represents the RMS transverse beam size (for KV beams, the total transverse beam size is $2{\sigma }_{x\mathrm{,}y}$). The derivatives σ_px,py are the conjugate variables $\text{d}{\sigma }_{x\mathrm{,}y}/\text{d}s={\sigma }_{\text{p}x\mathrm{,}\text{p}y}$. ${\kappa }_{x\mathrm{,0}}$ and ${\kappa }_{y\mathrm{,0}}$ are the external transverse focusing gradients in x and y, respectively. εx and εy are the transverse RMS emittances. K_sc is the space charge perveance defined by $K_{\text{sc}}=2N_{\text{L}}{r}_{\text{c}}/\left({\beta }^2{\gamma }^3\right)$, where N_L is the number of particles per unit length, r_c is the classical proton radius, and β and γ are relativistic factors. The RMS envelope equations can be derived from the envelope Hamiltonian in Eq. (1): $\begin{array}{l}\frac{{\text{d}}^2{\sigma }_x}{\text{d}{s}^2}+{\kappa }_{x\mathrm{,0}}^2{\sigma }_x-\frac{K_{\text{sc}}}{2\left({\sigma }_x+{\sigma }_y\right)}-\frac{{\epsilon }_x^2}{{\sigma }_x^3}=\mathrm{0,}\\ \frac{{\text{d}}^2{\sigma }_y}{\text{d}{s}^2}+{\kappa }_{y\mathrm{,0}}^2{\sigma }_y-\frac{K_{\text{sc}}}{2\left({\sigma }_x+{\sigma }_y\right)}-\frac{{\epsilon }_y^2}{{\sigma }_y^3}=0.\end{array}$ (2) Under constant focusing, obtaining the matched RMS beam sizes σ_{x, m} and σ_{y, m} via the corresponding algebraic equation set is straightforward: $\begin{array}{l}{\kappa }_{x\mathrm{,0}}^2{\sigma }_{x\mathrm{,}\text{m}}-\frac{K_{\text{sc}}}{2\left({\sigma }_{x\mathrm{,}\text{m}}+{\sigma }_{y\mathrm{,}\text{m}}\right)}-\frac{{\epsilon }_x^2}{{\sigma }_{x\mathrm{,}\text{m}}^3}=\mathrm{0,}\\ {\kappa }_{y\mathrm{,0}}^2{\sigma }_{y\mathrm{,}\text{m}}-\frac{K_{\text{sc}}}{2\left({\sigma }_{x\mathrm{,}\text{m}}+{\sigma }_{y\mathrm{,}\text{m}}\right)}-\frac{{\epsilon }_y^2}{{\sigma }_{y\mathrm{,}\text{m}}^3}=0.\end{array}$ (3) Here, we use the subscript “m” to denote the matched case.

The envelope equations in Eq. (2), which are typically employed to describe the oscillatory motion of the beam core, can be converted into a dimensionless form with a set of dimensionless variables, defined by $\{\begin{array}{lll}{\widehat{\sigma }}_x\hfill & =\hfill & {\sigma }_x/{\sigma }_{x\mathrm{,}\text{m}}\hfill \\ {\widehat{\sigma }}_y\hfill & =\hfill & {\sigma }_y/{\sigma }_{x\mathrm{,}\text{m}}\hfill \\ {\widehat{\sigma }}_{\text{p}x}\hfill & =\hfill & {\sigma }_{\text{p}x}/\left({\kappa }_{x\mathrm{,0}}{\sigma }_{x\mathrm{,}\text{m}}\right)\hfill \\ {\widehat{\sigma }}_{\text{p}y}\hfill & =\hfill & {\sigma }_{\text{p}y}/\left({\kappa }_{x\mathrm{,0}}{\sigma }_{x\mathrm{,}\text{m}}\right)\hfill \\ \tau \hfill & =\hfill & {\kappa }_{x\mathrm{,0}}s\hfill \end{array}$ (4) and the dimensionless parameters $\{\begin{array}{lll}{\mu }_x\hfill & =\hfill & \text{Δ}{\kappa }_x^2/{\kappa }_{x\mathrm{,0}}^2\hfill \\ r\hfill & =\hfill & {\sigma }_{y\mathrm{,}\text{m}}/{\sigma }_{x\mathrm{,}\text{m}}\hfill \\ \eta \hfill & =\hfill & {\kappa }_{y\mathrm{,0}}/{\kappa }_{x\mathrm{,0}}\hfill \\ {\epsilon }_r\hfill & =\hfill & {\epsilon }_y/{\epsilon }_x\mathrm{.}\hfill \end{array}$ (5) Here, $\text{Δ}{\kappa }_x^2=K_{\text{sc}}/\left[2{\sigma }_{x\mathrm{,}\text{m}}\left({\sigma }_{x\mathrm{,}\text{m}}+{\sigma }_{y\mathrm{,}\text{m}}\right)\right]$ represents the space-charge tuning depression in the matched case. Clearly, for constant focusing, the dimensionless matched beam sizes ${\widehat{\sigma }}_{x\mathrm{,}\text{m}}=\mathrm{1,}{\widehat{\sigma }}_{y\mathrm{,}\text{m}}=r$.

The corresponding dimensionless form of the Hamiltonian in Eq. (1) and envelope equations in Eq. (2) can be respectively expressed as $\begin{array}{ll}{\widehat{H}}_{\text{env}}=\hfill & \frac{1}{2}\left({\widehat{\sigma }}_{\text{p}x}^2+{\widehat{\sigma }}_{\text{p}y}^2+{\widehat{\sigma }}_x^2+{\eta }^2{\widehat{\sigma }}_y^2\right)-{\mu }_x\left(1+r\right)\ln \left({\widehat{\sigma }}_x+{\widehat{\sigma }}_y\right)\hfill \\ \hfill & +\frac{1-{\mu }_x}{2{\widehat{\sigma }}_x^2}+\frac{{\epsilon }_{\text{r}}^2\left(1-{\mu }_x\right)}{2{\widehat{\sigma }}_y^2}\mathrm{,}\hfill \end{array}$ (6) and $\begin{array}{l}\frac{{\text{d}}^{\text{2}}{\widehat{\sigma }}_x}{\text{d}{\tau }^2}+{\widehat{\sigma }}_x-\frac{{\mu }_x\left(\text{1}+\text{r}\right)}{\left({\widehat{\sigma }}_x+{\widehat{\sigma }}_y\right)}-\frac{1-{\mu }_x}{{\widehat{\sigma }}_x^3}=\mathrm{0,}\\ \frac{{\text{d}}^{\text{2}}{\widehat{\sigma }}_y}{\text{d}{\tau }^2}+{\eta }^{\text{2}}{\widehat{\sigma }}_y-\frac{{\mu }_x\left(\text{1}+\text{r}\right)}{\left({\widehat{\sigma }}_x+{\widehat{\sigma }}_y\right)}-\frac{{\epsilon }_{\text{r}}^2\left(1-{\mu }_x\right)}{{\widehat{\sigma }}_y^3}=0.\end{array}$ (7) In practice, beams are not fully matched because of magnetic errors or the misalignment of lattice elements; hence, $\left\{{\widehat{\sigma }}_x\mathrm{,}{\widehat{\sigma }}_y\right\}$ differs slightly from the matched solution $\left\{{\widehat{\sigma }}_{x\mathrm{,}\text{m}}\mathrm{,}{\widehat{\sigma }}_{y\mathrm{,}\text{m}}\right\}$, which is referred to as a beam-coherent mismatch oscillation. In the PCM, such mismatch oscillations provide the energy transferred from the beam core to single particles via space charge, forming halo particles when resonance occurs. A mismatched beam travelling in a constant-focusing channel can often be expressed as small perturbations ($\xi \mathrm{,}\zeta \mathrm{,}{\xi }_{\text{p}}\mathrm{,}{\zeta }_{\text{p}}$) on the matched solutions: $\begin{array}{l}{\widehat{\sigma }}_x={\widehat{\sigma }}_{x\mathrm{,}\text{m}}+\xi =1+\xi \mathrm{,}\\ {\widehat{\sigma }}_y={\widehat{\sigma }}_{y\mathrm{,}\text{m}}+\zeta =r+\zeta \mathrm{,}\\ {\widehat{\sigma }}_{\text{p}x}={\widehat{\sigma }}_{\text{p}x\mathrm{,}\text{m}}+{\xi }_{\text{p}}\mathrm{,}\\ {\widehat{\sigma }}_{\text{p}y}={\widehat{\sigma }}_{\text{p}y\mathrm{,}\text{m}}+{\zeta }_{\text{p}}\mathrm{.}\end{array}$ (8) Substituting Eq. (8) into Eq. (6), we can obtain the Hamiltonian for an envelope with perturbations: $\begin{array}{ll}{\widehat{H}}_{\text{per}}=\hfill & \frac{1}{2}[{\left({\widehat{\sigma }}_{\text{p}x\mathrm{,}\text{m}}+{\xi }_p\right)}^2+{\left({\widehat{\sigma }}_{\text{p}y\mathrm{,}\text{m}}+{\zeta }_{\text{p}}\right)}^2\hfill \\ \hfill & +{\left(1+\xi \right)}^2+{\eta }^2{\left(r+\zeta \right)}^2]\hfill \\ \hfill & -{\mu }_x\left(1+r\right)\ln (1+r+\xi +\zeta )\hfill \\ \hfill & +\frac{1-{\mu }_x}{2{\left(1+\xi \right)}^2}+\frac{{\epsilon }_{\text{r}}^2\left(1-{\mu }_x\right)}{2{\left(r+\zeta \right)}^2}\mathrm{.}\hfill \end{array}$ (9) By performing Taylor expansion and maintaining the linear term, from Eq. (9), we can obtain the equations of motion for the envelope perturbations in matrix form (further discussion in Appendix A): $\frac{{\text{d}}^2}{\text{d}{\tau }^2}\left(\begin{array}{c}\xi \\ \xi \end{array}\right)=-\left(\begin{array}{cc}{a}_0& a_1\\ a_1& a_2\end{array}\right)\left(\begin{array}{c}\xi \\ \xi \end{array}\right)$ (10) with the coefficients $\begin{array}{l}{a}_0=4\left(1-{\mu }_x\right)+\frac{r+2}{r+1}{\mu }_x\mathrm{,}\\ a_1=\frac{{\mu }_x}{r+1}\mathrm{,}\\ a_2=4\left({\eta }^2-\frac{{\mu }_x}{r}\right)+\frac{2r+1}{r+1}\frac{{\mu }_x}{r}\mathrm{.}\end{array}$ (11) As the coefficient matrix in Eq. (10) is symmetric, it can be decomposed via $A=U\cdot \text{diag}\left\{k_{\text{b}}^2\mathrm{,}{k}_{\text{q}}^2\right\}\cdot U^{\text{T}}$, where U is the eigenvector matrix of A. Here, k_b and k_q represent the wavenumbers of the “breathing mode” and “quadrupole mode,” respectively. The general solutions to Eq. (10) can be expressed as $\begin{array}{l}\xi (\tau )=C_{11}\cos \left(k_{\text{b}}\tau \right)+C_{12}\cos \left(k_{\text{q}}\tau \right)\mathrm{,}\\ \zeta (\tau )=C_{21}\cos \left(k_{\text{b}}\tau \right)+C_{22}\cos \left(k_{\text{q}}\tau \right)\mathrm{,}\end{array}$ (12) with the coefficients $C_{ij}={\displaystyle {\sum }_{k=1}^2{U}_{ij}}{U}_{jk}^T{\alpha }_k\mathrm{,}$ (13) where the two initial values ${\alpha }_1=\xi (0)\mathrm{,}{\alpha }_2=\zeta (0)$. Here, we use τ defined in Eq. (4) as the independent variable, and the mismatch of the monument does not consider (${\xi }_p\left(0\right)={\zeta }_p\left(0\right)=0$) such that Eq. (12) does not include the “sin” term. The envelope oscillation patterns depend on the initial values. For example, a pure breathing mode exists with $\zeta (0)=-\left(U_{21}/U_{22}\right)\xi (0)$ and a pure quadrupole mode with $\zeta (0)=-\left(U_{11}/U_{12}\right)\xi (0)$. Generally, beam-core oscillations can be characterized by a superposition of the two modes.

Single-particle motion with space charge of beam cores

In the presence of space charge, the horizontal motion of a test particle is governed by $\frac{{\text{d}}^{2}x}{\text{d}{s}^2}+{\kappa }_{x\mathrm{,0}}^{2}x=\frac{K_{\text{sc}}}{2{\sigma }_x\left({\sigma }_x+{\sigma }_y\right)}x$ (14) when the particle is inside the beam core ($\left|x\right|<2{\sigma }_x$), and $\frac{{\text{d}}^{2}x}{\text{d}{s}^2}+{\kappa }_{x\mathrm{,0}}^{2}x=\frac{2K_{\text{sc}}}{x^2+\left|x\right|\sqrt{x^2+4\left({\sigma }_y^2-{\sigma }_x^2\right)}}x$ (15) when the particle is outside the beam core ($\left|x\right|>2{\sigma }_x$). Here, x is the horizontal displacement from the center of the beam core. Using the dimensionless parameters defined in Eqs. (4) and (5), we can convert the equation of particle motion in Eqs. (14) and (15) into $\frac{{\text{d}}^2\widehat{x}}{\text{d}{\tau }^2}+\widehat{x}=\frac{{\mu }_x\left(1+r\right)}{{\widehat{\sigma }}_x\left({\widehat{\sigma }}_x+{\widehat{\sigma }}_y\right)}\widehat{x}\mathrm{,}\left|\widehat{x}\right|<{\widehat{\sigma }}_x\mathrm{,}$ (16) and $\frac{{\text{d}}^2\widehat{x}}{\text{d}{\tau }^2}+\widehat{x}=\frac{{\mu }_x\left(1+r\right)}{{\widehat{x}}^2+\left|\widehat{x}\right|\sqrt{{\widehat{x}}^2+\left({\widehat{\sigma }}_y^2-{\widehat{\sigma }}_x^2\right)}}\widehat{x}\mathrm{,}\left|\widehat{x}\right|>{\widehat{\sigma }}_x\mathrm{.}$ (17) Here, the dimensionless horizontal displacement is defined as $\widehat{x}=x/2{\sigma }_{x\mathrm{,}\text{m}}$. Eqs. (16) and (17) show that when the test particle travels inside and outside the beam core, the wavenumbers of the particles are different because of the varying space charge strength.

For particles inside the beam core, the wavenumber becomes minimum: Substituting Eqs. (8) and (12) into Eq. (16), we obtain $\frac{{\text{d}}^2\widehat{x}}{\text{d}{\tau }^2}+\left(1-{\mu }_x\right)\widehat{x}=f(\tau )\mathrm{,}$ (18) with minimum number of waves $k_{\text{p}\mathrm{,}\text{min}}=\sqrt{1-{\mu }_x}\mathrm{,}$ (19) where $f(\tau )$ is a function that describes the oscillation of the beam core. $\begin{array}{ll}f(\tau )=\hfill & -\frac{{\mu }_x}{1+r}\{\left[(2+r)C_{11}+C_{21}\right]\cos \left(k_{\text{b}}\tau \right)\hfill \\ \hfill & +\left[(2+r)C_{12}+C_{22}\right]\cos \left(k_{\text{q}}\tau \right)\}\mathrm{.}\hfill \end{array}$ (20) We can observe that k_p,min depends only on the space charge depression μ_x.

However, when the test particle is far from the beam core, the space charge can be neglected (μ_x = 0), and Eq. (17) becomes $\frac{{\text{d}}^2\widehat{x}}{\text{d}{\tau }^2}+\widehat{x}=\mathrm{0,}$ (21) At this zero-space-charge limit, the wavenumber reaches its maximum, $k_{\text{p}\mathrm{,}\text{max}}=1.$ (22) Generally, for a single particle travelling outside the beam core, $\sqrt{1-{\mu }_x}\le k_{\text{p}}\le 1$.

In the PCM, particles move periodically inside and outside the beam core. In particular, when the wavenumbers of a test particle and beam core satisfy $k_{\text{p}}=\frac{k_{\text{b,q}}}{2}\mathrm{,}$ (23) where k_b and k_q are the wavenumbers of the “breathing mode” and “quadrupole mode” the 2:1 parametric resonance occurs and forms a beam halo [37].

To illustrate this, we plot the wavenumbers as functions of the beam current (in units of the normalized space-charge tune depression μx defined in Eq. 5) by solving Eqs. (16) and (17) for two representative cases: round and elliptical beams. In Fig. 1, the blue dashed lines represent the two limits of the single-particle wavenumber $k_{\text{p}\mathrm{,}\text{min}}=\sqrt{1-{\mu }_x}$ and k_p,max = 1), whereas the solid lines represent half of the wavenumbers for breathing mode k_b/2 and quadrupole mode k_q/2.

Fig. 1Full-screen View

(Color online) Half wavenumber of the breathing mode k_b/2 (in red) and the quadrupole mode k_q/2 (in yellow) versus the beam current in the unit of μ_x for round (upper panel) and elliptical beams (lower panel). The maximum (k_p,max) and minimum (k_p,min) wave-numbers of single particles are also plotted for comparison

Figure 1a shows that for round beams, k_p,min < k_q/2, k_b/2 < k_p,max always holds as the beam current increases from the zero space charge limit (μ_x = 0) to the extreme space charge limit (μ_x = 1.0), indicating that both envelope modes can resonate with the test particle and drive the 2:1 parametric resonance. In comparison, for the elliptical beams shown in Fig. 1b, we obtain k_p,min < k_q/2, k_b/2 < k_p,max when μ_x > 0.39, which implies that the breathing mode can drive the 2:1 parametric resonance in the range of μ_x > 0.39. For μ_x < 0.39, we have k_p,min < k_q/2 < k_p,max and k_b/2 > k_p,max. In this case, the 2:1 parametric resonance can only be driven by the quadrupole mode.

Round beams in the breathing mode

Let us consider a round beam traveling in a symmetric focusing channel with initial equal mismatch perturbations on x and y, for example ξ(0) = ξ(0) = 0.05, to excite a breathing mode on the beam (i.e., an in-phase pattern and 5% mismatch; the quadruple mode is absent here). We select four single particles with different initial (dimensionless) horizontal displacements, ${\widehat{x}}_1<1.0<{\widehat{x}}_2<{\widehat{x}}_3<{\widehat{x}}_4$ (shown in Fig. 2; the dimensionless beam-core radius is 1), and the momentum is zero. By numerically solving Eqs. (16) and (17), we plot the particle motion is on a Poincaré map for the two representative cases in Fig. 2. The moderate space charge μ_x = 0.1 and strong space charge x = 0.8 are discussed in detail below.

Fig. 2Full-screen View

(Color online) Motion of single particles with different initial positions in the presence of the breathing mode with moderate (μ_x = 0.1, left column a, b and c) and strong space charge (μ_x = 0.8, right column d, e and f). The Poincaré sections are shown in a and d; the wavenumber of single particle (normalized to the breathing mode) as a function of initial positions is shown in b and e; and the maximum displacement of single particle as a function of initial displacement is shown in c and f compared with the simulation results (red dotted lines)

First, we analyze the moderate space charge case μ_x = 0.1. For the two test particles with initial positions ${\widehat{x}}_1$ and ${\widehat{x}}_2$, the motions of the particles take the form of regular ellipses on the Poincaré map in Fig. 2a. In comparison, the third particle with initial position ${\widehat{x}}_3$ experiences a large excursion (red trajectory), indicating that resonance appears. In this case, energy is transferred from the beam core to the particle, driving the particle to reach the largest displacement ${\widehat{x}}_{3,\max }$.

The “lock” of the wavenumber of the third particle in Fig. 2b is a typical characteristic of the parametric resonance. Within the resonance island (red contour in Fig. 2a), the wavenumber is equal to half of the wavenumber of the breathing mode (k = k_b/2) in Fig. 2b. In comparison, the wavenumber increases outside the resonance region with a larger initial position (${\widehat{x}}_4$ in Fig. 2a, for example) when parametric resonance is absent. The lock on the half wavenumber indicates the occurrence of a 2:1 parametric resonance.

PIC simulations were conducted using the PyORBIT code to support the above numerical results and analysis. In the simulation, a constant-focusing channel with 16 “equal cells” was employed. The main parameters used in the simulation are summarized in Table 1. The space charge solver is based on the fast Fourier transform (FFT) method [44]. The simulation results for the maximum displacement of the particles as a function of the initial displacement atμ_x = 0.1 are shown in Fig. 2c. The simulation results were in good agreement with the numerical solutions.

Second, with an enhanced space charge (μ_x = 0.8), the motion of single particles becomes more complicated, as shown in Fig. 2d–f (right). Compared with the moderate space charge case (μ_x = 0.1), three small islands are observed in addition to the 2:1 resonance island in Fig. 2d. A closer examination of Fig. 2e reveals that in the region of $1.080\times \widehat{x}<1.085$, the wavenumber is locked at k = (1/3)k_b, indicating a 3:1 parametric resonance. Compared with the 2:1 resonance case, the small areas of the three islands indicate that the 3:1 resonance is much weaker. (The corresponding narrow stopband is shown in purple in Fig. 2e.)

PIC simulations with μ_x = 0.8 were also performed. As shown in Fig. 2d, the particles were trapped within the 2:1 resonance island (shown as red dots). The purple dots around the three islands indicate that the particles gathered around the 3:1 resonance island because of the weak 3:1 resonance. The simulation results for the maximum particle displacement are shown in Fig. 2f. These results agreed with the numerical calculations.

An interesting question may be raised as to whether the 4:1 parametric resonance exists and contributes to the beam halo for a strong space charge withμ_x = 0.8. As shown in Fig. 3, we obtain k_b/4 < k_p,min when μ_x = 0.8, indicating that a 4:1 or higher parametric resonance cannot be excited. In comparison, the condition k_p,min < k_b/3 < 3k_b/8 < k_p,max is satisfied when μ_x = 0.8, which supports the occurrence of the 3:1 and 8:3 resonance islands shown in Fig. 2.

Fig. 3Full-screen View

(Color online) One third (k_b/3), one quarter (k_b/4), and 8:3 (3k_b/8) of the wavenumber of the breathing mode versus beam current in the unit of μ_x for round beams. The maximum (k_p,max) and minimum (k_p,min) wavenumber of single particles is also plotted for comparison

Round beams with mixed modes

In this subsection, we consider a round beam perturbed withξ(0) ≠ξ(0), which can simultaneously drive both the breathing and quadrupole modes. The motion of single particles affected by the space charge of the two modes is obtained and plotted on a Poincaré map in Fig. 4, which shows that both modes can drive the 2:1 parametric resonance when the wavenumber of the test particle k satisfies k = k_q/2 and k = k_b/2, respectively.

Fig. 4Full-screen View

(Color online) Motion of single particles with different initial positions in the presence of mixed mode with moderate (μ_x = 0.1, left column a, b, and c) and strong space charge (μ_x = 0.8, right column d, e, and f). The Poincaré sections are shown in a and d; the wavenumbers of single particles as functions of initial positions are shown in b and e; and the maximum displacements of single particles as functions of initial displacement are shown in c and f, compared with the simulation results (red dotted lines)

Furthermore, a chaotic phenomenon appears in the mixed modes, as indicated by the black region in Fig. 4a and d. In contrast to the characteristic “lock of the wave-number” of the parametric resonance, when chaos occurs, the wavenumber of the test particle is random, as shown in Fig. 4b and e.

PIC simulations were performed for the mixed modes, and the results are indicated in Fig. 4c and f using red dots. With a moderate space charge (μ_x = 0.1), the simulation results for the maximum displacements agreed with the numerical calculations.

Note that the disagreement between the simulation result and numerical calculation becomes observable in the strong space charge case, as shown in Figs. 2f and 4f. We attribute this to the disturbance of the uniform particle distribution during self-consistent particle tracking, resulting in different wavenumbers of single particles from the PCM.

Elliptical beams

In the following, we discuss the case of elliptical beams, i.e., r ≠ 1. The wavenumbers as functions of the normalized space charge tune depression μ_x are numerically obtained and shown in Fig. 1b, where η = 1.25 and r = 0.9. For a moderate space charge (μ_x = 0.1), only the quadrupole mode can induce the 2:1 resonance.

The motions of single particles affected by the space charge of the two modes in the elliptical beams are obtained and plotted on a Poincaré map in Fig. 5. The characteristic “lock of the wavenumber” is shown in Fig. 5b. For μ_x ≥ 0.5, two resonance islands appear, as shown in Fig. 5d and g, which are driven by the quadrupole and breathing modes, respectively. The locks of the two modes are shown in Fig. 5e and h. Furthermore, Fig. 5d and g shows that as space charge increases, the two resonance islands approach each other. With strong space charge (μ_x = 0.8), as shown quadrupole modes are closer to each other, which causes chaotic phenomena around the two adjacent resonant islands (shown as black dots).

Fig. 5Full-screen View

(Color online) Motion of single particles with different initial positions under the mixed mode with moderate (μ_x = 0.1, left column a, b and c), enhanced (μ_x = 0.5, center column d, e, and f) and strong space charge (μ_x = 0.8, right column g, h, and i). The Poincaré sections are shown in a, d, and g; the wavenumbers of single particles as functions of initial positions are shown in b, e, and h; and the maximum displacements of single particles as functions of initial displacement are shown in c, f, and i, compared with the simulation results (red dotted lines)

PIC simulations were performed for the mixed modes in elliptical beams using the parameters listed in Table 1. The simulation results for the maximum displacements are shown in Fig. 5c, f, and i using red points. The simulation results were in good agreement with the numerical calculations.

Generalized PCM with dispersion

For beams traveling in a constant-focusing bending channel, the transverse beam dynamics can be described by the envelope equation set with the dispersion function, given by [2, 51] $\begin{array}{l}\frac{{\text{d}}^2}{\text{d}{s}^2}{\sigma }_x+{\kappa }_{x\mathrm{,0}}^2{\sigma }_x-\frac{K_{\text{sc}}}{2X(X+Y)}{\sigma }_x-\frac{{\epsilon }_{\text{d}x}^2}{{\sigma }_x^3}=\mathrm{0,}\\ \frac{{\text{d}}^2}{\text{d}{s}^2}{\sigma }_y+{\kappa }_{y\mathrm{,0}}^2{\sigma }_y-\frac{K_{\text{sc}}}{2Y(X+Y)}{\sigma }_y-\frac{{\epsilon }_{\text{d}y}^2}{{\sigma }_y^3}=\mathrm{0,}\\ \frac{{\text{d}}^2}{\text{d}{s}^2}{D}_{\delta }+{\kappa }_{x\mathrm{,0}}^2{D}_{\delta }-\frac{K_{\text{sc}}}{2X(X+Y)}{D}_{\delta }=\frac{{\sigma }_p}{\rho }\mathrm{,}\end{array}$ (24) which can be derived from the dispersion-modified envelope Hamiltonian: $H_{\text{env,d}}=\frac{1}{2}\left({\sigma }_{\text{p}x}^2+{\sigma }_{\text{p}y}^2+D_{\text{p}\delta }^2\right)+V_{\text{env,d}}\left({\sigma }_x\mathrm{,}{\sigma }_y\mathrm{,}{D}_{\delta }\right)$ (25) with $\begin{array}{ll}{V}_{\text{env,d}}\left({\sigma }_x\mathrm{,}{\sigma }_y\mathrm{,}{D}_{\delta }\right)\hfill & =\frac{1}{2}{\kappa }_{x\mathrm{,0}}^2{\sigma }_x^2+\frac{1}{2}{\kappa }_{y\mathrm{,0}}^2{\sigma }_y^2\hfill \\ \hfill & -\frac{K_{\text{sc}}}{2}\ln (X+Y)\hfill \\ \hfill & +\frac{{\epsilon }_{\text{d}x}^2}{2{\sigma }_x^2}+\frac{{\epsilon }_{\text{d}y}^2}{2{\sigma }_y^2}-\frac{{\sigma }_{\text{p}}{D}_{\delta }}{\rho }\mathrm{.}\hfill \end{array}$ (26) Here, $X=\sqrt{{\sigma }_x^2+D_{\delta }^2}$ is the total RMS horizontal beam size, whre σx is the betatron beam size, and Dδ is the “dispersion beam size,” defined as $D_{\delta }\equiv D_x{\sigma }_{\text{p}}$. $D_{\text{p}\delta }$ is the derivative of Dδ, with $D_{\text{p}\delta }\equiv \text{d}{D}_{\delta }/\text{d}s$. $Y={\sigma }_y$ is the RMS vertical beam size because the dispersion effect is considered here only on x. The subscript “d” denotes the case with dispersion. A matched equation set with dispersion is $\begin{array}{l}{\kappa }_{x\mathrm{,0}}^2{\sigma }_{x\mathrm{,}\text{m}}-\frac{K_{\text{sc}}}{2X_{\text{m}}\left(X_{\text{m}}+Y_{\text{m}}\right)}{\sigma }_{x\mathrm{,}\text{m}}-\frac{{\epsilon }_{\text{d}x}^2}{{\sigma }_x^3}=\mathrm{0,}\\ {\kappa }_{y\mathrm{,0}}^2{\sigma }_{y\mathrm{,}\text{m}}-\frac{K_{\text{sc}}}{2Y_{\text{m}}\left(X_{\text{m}}+Y_{\text{m}}\right)}{\sigma }_{y\mathrm{,}\text{m}}-\frac{{\epsilon }_{\text{d}y}^2}{{\sigma }_y^3}=\mathrm{0,}\\ {\kappa }_{x\mathrm{,0}}^2{D}_{\delta \mathrm{,}\text{m}}-\frac{K_{\text{sc}}}{2X_{\text{m}}\left(X_{\text{m}}+Y_{\text{m}}\right)}{D}_{\delta \mathrm{,}\text{m}}=\frac{{\sigma }_{\text{p}}}{\rho }\mathrm{,}\end{array}$ (27) where $X_{\text{m}}$, $Y_{\text{m}}$, and $D_{\delta \mathrm{,}\text{m}}$ are the matched solutions.

Based on the dimensionless parameters defined in Eq. (4), the Hamiltonian in Eq. (25) can be rewritten as $\begin{array}{ll}{\widehat{H}}_{\text{d}\mathrm{,}\text{env}}=\hfill & \frac{1}{2}\left({\widehat{\sigma }}_{\text{p}x}^2+{\widehat{\sigma }}_{\text{p}y}^2+{\widehat{D}}_{\text{p}\delta }^2\right)+\frac{1}{2}\left({\widehat{\sigma }}_x^2+{\eta }^2{\widehat{\sigma }}_y^2+{\widehat{D}}_{\delta }^2\right)\hfill \\ \hfill & -\frac{{\mu }_{x\mathrm{,}\text{d}}\left(\text{1}+\text{R}\right)}{\sin {\theta }^2}\ln (\widehat{X}+\widehat{Y})+\frac{1-{\mu }_{x\mathrm{,}\text{d}}}{2}\frac{1}{{\widehat{\sigma }}_x^2}\hfill \\ \hfill & +\frac{{\epsilon }_{r\mathrm{,}\text{d}}^2\left(1-{\mu }_{x\mathrm{,}\text{d}}\right)}{2}\frac{1}{{\widehat{\sigma }}_y^2}+\left(1-{\mu }_{x\mathrm{,}\text{d}}\right)\frac{\cos \theta }{\sin \theta }{\widehat{D}}_{\delta }\mathrm{,}\hfill \end{array}$ (28) where ${\widehat{\sigma }}_x$ and ${\widehat{\sigma }}_y$ are the normalized betatron beam sizes, and η is the focusing ratio defined in Eq. (5). The dimensionless variables related to the dispersion in Eq. (28) are defined as $\{\begin{array}{lll}\widehat{X}\hfill & =\hfill & X/{\sigma }_{x\mathrm{,}\text{m}}\hfill \\ \widehat{Y}\hfill & =\hfill & Y/{\sigma }_{x\mathrm{,}\text{m}}\hfill \\ {\widehat{D}}_{\delta }\hfill & =\hfill & D_{\delta }/{\sigma }_{x\mathrm{,}\text{m}}\hfill \\ R\hfill & =\hfill & Y_{\text{m}}/X_{\text{m}}\hfill \\ \sin \theta \hfill & =\hfill & {\sigma }_{x\mathrm{,}\text{m}}/X_{\text{m}}\hfill \\ \cos \theta \hfill & =\hfill & D_{\delta \mathrm{,}\text{m}}/X_{\text{m}}\hfill \end{array}$ (29) and $\{\begin{array}{l}{\epsilon }_{r\mathrm{,}\text{d}}={\epsilon }_{\text{d}y}/{\epsilon }_{\text{d}x}\hfill \\ {\mu }_{x\mathrm{,}\text{d}}=\text{Δ}{\kappa }_{x\mathrm{,}d}^2/{\kappa }_{x\mathrm{,0}}^2\mathrm{.}\hfill \end{array}$ (30) Here, $\sin \theta ={\sigma }_{x\mathrm{,}\text{m}}/X_{\text{m}}$ is the betatron ratio, and $\cos \theta =D_{\delta \mathrm{,}\text{m}}/X_{\text{m}}$ is the dispersion ratio. $\text{Δ}{\kappa }_{x\mathrm{,}\text{d}}^2=K_{\text{sc}}/\left[2X_{\text{m}}\left(X_{\text{m}}+{\sigma }_{y\mathrm{,}\text{m}}\right)\right]$ represents the space-charge tune depression in the presence of dispersion.

We further introduce the “dispersion strength” $\text{Λ}\equiv D_{\delta \mathrm{,}\text{m}}^{\left(0\right)}/{\sigma }_{x\mathrm{,}\text{m}}^{\left(0\right)}$ to characterize the ratio of dispersion motion to betatron motion in zero-current beam case. Here, the superscript “(0)” denotes the absence of space charge, and ${\sigma }_{x\mathrm{,}\text{m}}={\sigma }_{x\mathrm{,}\text{m}}^{\left(0\right)}/\sqrt[4]{1-{\mu }_x}$ and $D_{\delta \mathrm{,}\text{m}}=D_{\delta \mathrm{,}\text{m}}^{\left(0\right)}/\left(1-{\mu }_x\right)$. For typical synchrotrons such as the CSNS RCS [52, 53], the energy spread is frequently less than 1%, and we obtain 0 < Λ < 1.0.

The dimensionless form of the dispersion-modified envelope equation set can be obtained from the Hamiltonian in Eq. (28) as follows: $\begin{array}{l}\frac{{\text{d}}^2}{\text{d}{\tau }^2}{\widehat{\sigma }}_x+{\widehat{\sigma }}_x-\frac{{\mu }_{x\mathrm{,}\text{d}}\left(1+R\right){\widehat{\sigma }}_x}{\widehat{X}(\widehat{X}+\widehat{Y})\sin {\theta }^2}-\frac{1-{\mu }_{x\mathrm{,}\text{d}}}{{\widehat{\sigma }}_x^3}=\mathrm{0,}\\ \frac{{\text{d}}^2}{\text{d}{\tau }^2}{\widehat{\sigma }}_y+{\eta }^2{\widehat{\sigma }}_y-\frac{{\mu }_{x\mathrm{,}\text{d}}\left(\text{1}+\text{R}\right){\widehat{\sigma }}_{\text{y}}}{\widehat{Y}(\widehat{X}+\widehat{Y})\sin {\theta }^2}-\frac{{\epsilon }_{\text{r}\mathrm{,}\text{d}}^2\left(1-{\mu }_{x\mathrm{,}\text{d}}\right)}{{\widehat{\sigma }}_y^3}=\mathrm{0,}\\ \frac{{\text{d}}^2}{\text{d}{\tau }^2}{\widehat{D}}_{\delta }+{\widehat{D}}_{\delta }-\frac{{\mu }_{x\mathrm{,}\text{d}}\left(\text{1}+\text{R}\right){\widehat{D}}_{\delta }}{\widehat{X}(\widehat{X}+\widehat{Y})\sin {\theta }^2}-\left(1-{\mu }_{x\mathrm{,}\text{d}}\right)\frac{\cos \theta }{\sin \theta }=0.\end{array}$ (31) From Eq. (31), we obtain the matched beam sizes ${\widehat{\sigma }}_{x\mathrm{,}\text{m}}=1$, ${\widehat{\sigma }}_{y\mathrm{,}\text{m}}=r$, ${\widehat{D}}_{\delta \mathrm{,}\text{m}}=\cot \theta$ and ${\widehat{X}}_{\text{m}}=\csc \theta$. Substituting the mismatch perturbations $\begin{array}{c}{\widehat{\sigma }}_x=1+\xi \\ {\widehat{\sigma }}_y=r+\zeta \\ {\widehat{D}}_{\delta }=\cot \theta +d\end{array}$ (32) into the dispersion-modified envelope set of Eq. (31) (“d” is the perturbation on the dispersion), we obtain (more details in Appendix B) $\frac{{\text{d}}^2}{\text{d}{\tau }^2}\left(\begin{array}{c}\xi \\ \xi \\ d\end{array}\right)=-\left(\begin{array}{ccc}{b}_0& b_1& b_2\\ b_1& b_3& b_4\\ b_2& b_4& b_5\end{array}\right)\left(\begin{array}{c}\xi \\ \xi \\ d\end{array}\right)$ (33) with the coefficients $\begin{array}{l}{b}_0=4\left(1-{\mu }_{x\mathrm{,}\text{d}}\right)+\frac{R+2}{R+1}{\mu }_{x\mathrm{,}\text{d}}\sin {\theta }^2\mathrm{,}\\ b_1=\frac{1}{R+1}{\mu }_{x\mathrm{,}\text{d}}\sin \theta \mathrm{,}\\ b_2=\frac{R+2}{R+1}{\mu }_{x\mathrm{,}\text{d}}\sin \theta \cos \theta \mathrm{,}\\ b_3=4\left({\eta }^2-\frac{{\mu }_{x\mathrm{,}\text{d}}}{R}\right)+\frac{2R+1}{R+1}\frac{{\mu }_{x\mathrm{,}\text{d}}}{R}\mathrm{,}\\ b_4=\frac{1}{R+1}{\mu }_{x\mathrm{,}\text{d}}\cos \theta \mathrm{,}\\ b_5=\left(1-{\mu }_{x\mathrm{,}\text{d}}\right)+\frac{R+2}{R+1}{\mu }_{x\mathrm{,}\text{d}}\cos {\theta }^2\mathrm{.}\end{array}$ (34) The coefficients in Eq. (34) can be expressed as a real symmetric matrix $B=\left\{b_{ij}\right\}$. Similar to the treatment described in Sec. 2, we obtain $B=U_{\text{d}}\cdot \text{diag}\left\{k_{\text{b}}^2\mathrm{,}{k}_{\text{q}}^2\mathrm{,}{k}_{\text{d}}^2\right\}\cdot U_d^T$. Compared with Eq. (10), the dispersion mode kd can be identified [2]. The general solution to Eq. (33) takes the form $\begin{array}{l}\xi =D_{11}\cos \left(k_{\text{b}}\tau \right)+D_{12}\cos \left(k_{\text{q}}\tau \right)+D_{13}\cos \left(k_{\text{d}}\tau \right)\mathrm{,}\\ \zeta =D_{21}\cos \left(k_{\text{b}}\tau \right)+D_{22}\cos \left(k_{\text{q}}\tau \right)+D_{23}\cos \left(k_{\text{d}}\tau \right)\mathrm{,}\\ d=D_{31}\cos \left(k_{\text{b}}\tau \right)+D_{32}\cos \left(k_{\text{q}}\tau \right)+D_{33}\cos \left(k_{\text{d}}\tau \right)\mathrm{.}\end{array}$ (35) Here $D_{ij}={\displaystyle {\sum }_{k=1}^3{U}_{\text{d}\mathrm{,}ij}}{U}_{\text{d}\mathrm{,}jk}^T{\alpha }_k$, with ${\alpha }_1=\xi (0)\mathrm{,}{\alpha }_2=\zeta (0)\mathrm{,}{\alpha }_3=d(0)$.

However, in the presence of space charge and dispersion, the motion of a single particle is governed by $\begin{array}{ll}\frac{{\text{d}}^2{\widehat{x}}_{\text{d}}}{\text{d}{\tau }^2}+{\widehat{x}}_{\text{d}}\hfill & =\frac{{\mu }_{\text{d}\mathrm{,}x}\left(1+R\right)}{\sin {\theta }^2}\frac{{\widehat{x}}_{\text{d}}}{\widehat{X}(\widehat{X}+\widehat{Y})}\hfill \\ \hfill & +\frac{\widehat{\delta }}{2}\left(1-{\mu }_{\text{d}\mathrm{,}x}\right)\cos \theta \hfill \end{array}$ (36) when the particle is inside the beam core ($\left|{\widehat{x}}_{\text{d}}\right|<\widehat{X}\sin \theta$) and $\begin{array}{ll}\frac{{\text{d}}^2{\widehat{x}}_{\text{d}}}{\text{d}{\tau }^2}+{\widehat{x}}_{\text{d}}\hfill & =\frac{{\mu }_{\text{d}\mathrm{,}x}\left(1+R\right){\widehat{x}}_{\text{d}}}{{\widehat{x}}_{\text{d}}^2+\left|{\widehat{x}}_{\text{d}}\right|\sqrt{{\widehat{x}}_{\text{d}}^2+\left({\widehat{Y}}^2-{\widehat{X}}^2\right)\sin {\theta }^2}}\hfill \\ \hfill & +\frac{\widehat{\delta }}{2}\left(1-{\mu }_{\text{d}\mathrm{,}x}\right)\cos \theta \hfill \end{array}$ (37) when the particle is outside the beam core ($\left|{\widehat{x}}_{\text{d}}\right|>\widehat{X}\sin \theta$). Here, ${\widehat{x}}_{\text{d}}=x/\left(2X_{\text{m}}\right)$, and $\widehat{\delta }=\delta /{\sigma }_p$ denotes the ratio of the momentum spread between the single particle and beam core. Clearly, for σp = 0, Λ = 0 and $\sin \theta =1$, R = r. In this case, Eqs. (36) and (37) are equivalent to Eqs. (16) and (17).

An interesting question may be raised: Can the dispersion mode excite the 2:1 parametric resonance and induce a beam halo, as in the case of the two envelope modes discussed in Sec. 3? To analyze this problem, we plot the (half) wavenumber of the dispersion mode and the two envelope modes in Fig. 6, where the corresponding parameters are Λ = 0.2, η = 1.0, R = 0.978 and Λ = 1.0, η = 1.0, R = 0.688. Figure 6 shows that the half wavenumber of the dispersion mode is always below the lower limit of the test particle, indicating that the dispersion mode cannot induce a 2:1 resonance. Moreover, by inspecting the figure, we observe k_,min < k_d < k_p,max, implying the dispersion mode can generate a “1:1” parametric resonance on single particles.

Fig. 6Full-screen View

(Color online) Half wavenumber of the breathing mode k_b/2 (in red), quadrupole mode k_q/2 (yellow), dispersion mode k_d/2 (green), and wavenumber of the dispersion mode k_d (black) versus beam current in the unit of μx in the presence of dispersion with Λ = 0.2 (upper) and Λ = 1.0 (lower). The maximum (k_p,max) and minimum (${\displaystyle {\sum }_a^b{k}_{\text{p},\min }}$) wavenumbers of single particles are also plotted for comparison

In the following, we first demonstrate the mechanism of the 1:1 parametric resonance under the combined effect of dispersion and space charge and then show the damping effect on the beam halo formation owing to the moving stably fixed point (SFP) of single particles.

Dispersion-induced 1:1 parametric resonance

First, we consider the case of pure modes, in which only one of the breathing, quadrupole, or dispersion modes exists in the beam-core oscillation pattern. This can be achieved by setting the coefficients Dij = 0 in Eq. (35).

The parametric resonances for the beam halo formation in the presence of space charge and dispersion can be analyzed using the dispersion-modified PCM in Eqs. (31), (36), and (37). As shown in Fig. (7), two resonant islands are observed in panels (a) and (b), indicating the occurrence of a 2:1 resonance driven by the (pure) breathing and (pure) quadrupole modes. In the presence of the dispersion mode, a “crescent moon” island is observed, as shown in panel (c). Compared with the resonant islands in panels (a) and (b), only one island exists in panel (c), which is identified as the 1:1 resonance induced by the dispersion mode.

Fig. 7Full-screen View

(Color online) Poincaré sections of single particles with $\widehat{\delta }=3.0$ affected by the breathing mode (a), quadrupole mode (b), dispersion mode (c) and mixed modes (d), with Λ = 0.2 and μx =0.1. The wavenumbers of single particles for mixed modes as functions of initial positions are shown in (e); the maximum displacement of single particle as a function of initial displacement are shown in (f), compared with the simulation results (red dotted lines)

To better illustrate the 1:1 resonance, we plot the Poincaré section and normalized wavenumber of single particles in Fig. 8 (i.e., enlarged plot in Fig. 7(c)). We observe that in the “crescent moon” island, the wavenumber of single particle is locked and equal to wavenumber of the dispersion mode (marked as the yellow shaded area in Fig. 8(b)). This is a clear proof to support the “crescent moon” island in the Poincaré section is a 1:1 parametric resonance driven by the dispersion mode.

Fig. 8Full-screen View

(Color online) Poincaré section of single particles with $\widehat{\delta }=3.0$ affected by the dispersion mode (a) with Λ = 0.2 and μx =0.1; the wavenumber (normalized to the dispersion mode) of single particles as a function of initial positions are shown in (b)

Next, we analyze the mixed mode, in which the breathing, quadrupole, and dispersion modes exist simultaneously. As shown in Fig. 7(d), driven by the mixed modes, the 2:1 resonance driven by the two envelope modes and the 1:1 resonance induced by the dispersion mode exist. Furthermore, chaos is observed around the resonant islands. Panel (e) presents the corresponding wavenumbers of the three modes. Similar to the results shown in Fig. 4 in Sect. 3, the locking of the wavenumbers indicates the 2:1 and 1:1 parametric resonances, which agrees with the resonant islands in panel (d). When chaos occurs, the wavenumber becomes random.

PIC simulations were conducted using the parameters of the Λ=0.2 case, as shown in Table 2. The simulation results are shown in Fig. 7(f). The simulation results of maximum displacements were in good agreement with the numerical calculations using the generalized PCM with dispersion.

Alleviation of the beam halo using strong dispersion

4.3.1

Motion of single particles with zero momentum deviation

We assume single particles with zero momentum deviation for the synchronous particles ($\widehat{\delta }=0$) in high-intensity synchrotrons. As shown in Fig. 6, for a large dispersion (Λ = 1.0), k_q/2 < k_p,min holds when μx < 0.66, indicating that the quadrupole mode cannot excite the 2:1 resonance in this region. In comparison, for the breathing mode, we obtain k_b/2 > k_p,min and thus can trigger the 2:1 resonance.

The Poincaré sections of single particles with $\widehat{\delta }=0$ for the pure mode case with different dispersion strengths are shown in Fig. 9. Figure 9 (a) and (d) show that the areas of the 2:1 resonance islands driven by the breathing mode decrease with increasing dispersion strength. A similar phenomenon is observed in Fig. 9 (b) and (e): with an enhanced dispersion strength Λ =1.0, the 2:1 resonance driven by the quadrupole mode disappears. We attribute this result to the fact that the dispersion effect can reduce the 2:1 resonance. Second, no resonance islands are observed in Fig. 9(c) and (f), implying that the dispersion mode cannot induce a 2:1 resonance, which agrees with the analysis from Fig. 6.

Fig. 9Full-screen View

(Color online) Poincaré section of single particles with $\widehat{\delta }=0.0$ affected by the breathing mode (a) and (d), the quadrupole mode (b) and (e), and the dispersion mode (c) and (f), with Λ = 0.2 (left column) and Λ = 1.0 (right column) under the fixed space charge strength μx =0.1

The Poincaré sections for the mixed mode case are shown in Fig. 10. The chaos existed around the resonance islands driven by mixed modes. Compared with Fig. 10(a), we observe the chaos in Fig. 10(d) weakens as the dispersion strength increases (Λ = 1.0). The wavenumbers of single particles with different initial positions are shown in in Fig. 10(b) and (e), where the red and gray shadows represent the resonance and chaos, respectively. PIC simulations were performed for Λ = 0.2 and Λ = 1.0 using the parameters listed in Table 2. The simulation results for the maximum displacements are shown in Fig. 10(c) and (f). The simulation results were in good agreement with the PCM calculations.

Fig. 10Full-screen View

(Color online) Motion of zero-momentum-deviation single particles ($\widehat{\delta }=0.0$) with different initial positions in the presence of mixed mode with moderate (Λ = 0.2, left column (a), (b) and (c)) and strong dispersion (Λ = 1.0, right column (d), (e) and (f)) under the fixed space charge tune depression μx = 0.1. The Poincaré sections are shown in (a) and (d); the wavenumber of single particle as a function of initial positions are shown in (b) and (e); the maximum displacement of single particle as a function of initial displacement are shown in (c) and (f), compared with the simulation results (red dotted lines)

4.3.2

Motion of single particles with large momentum deviation

For the single particles with large momentum deviation, the SFP can “move” out of the beam core in the presence of the dispersion effect. As an example, we plot four SFPs with different momentum deviations in phase space in the panel (a) of Fig. 11. We set Λ = 1.0 for the beam core. We observe that with $\widehat{\delta }=3.0$ and 4.0 (marked as S₃ and S₄), the two SFPs are located outside the beam core.

Fig. 11Full-screen View

(Color online) (a) Example with Λ=1.0, μx = 0.1 of the moving SFP at four different positions (inside the beam core S1, S₂, and outside the beam core S₃, S₄) with increasing momentum deviations of single particles ($\widehat{\delta }=\mathrm{1.0,2.0,3.0,4.0}$); (b) example with Λ=1.0, μx = 0.1 of four single particles with their initial positions at the edge of the beam core and different momentum deviations $\widehat{\delta }=\mathrm{1.0,2.0,3.0,4.0}$. The trajectories of the two particles inside the beam core (in red and green), with their maximum displacements M₁ and M₂ overlapping. The trajectories of the other two particles outside the beam core (in yellow and blue), with their maximum displacements M₃ and M₄

For a more illustrative discussion, the scanning of the maximum excursion of an edge particle is shown in Fig. 12. The scan is performed by varying the dispersion strength of the beam core Λ and the momentum deviation ratio of the single particle $\widehat{\delta }$. Here, the edge particles are defined that their initial positions are set to “sit” on the edge of the beam core, i.e., ${\widehat{x}}_{\text{d}}\left(0\right)=1$. For different $\widehat{\delta }$ and Λ values, the edge particles have different maximum excursions, ${\widehat{x}}_{\text{max}}$. The advantage of using edge particles is that, (shown in Fig. 11(b)), for the edge particles with ${\widehat{x}}_{\text{max}}=1$, the SFPs are located inside the beam core, shown as the region “inner SFP” in Fig. 12; for the edge particles with ${\widehat{x}}_{\text{max}}>1$, shown as the region “outer SFP” in Fig. 12.

Fig. 12Full-screen View

(Color online) Scan of the maximum motion of the “edge particles” (${\widehat{x}}_{\text{max}}$) with varying momentum deviations ($\widehat{\delta }$) and dispersion strength (Λ), under the fixed space-charge tune depression μx = 0.1. Black line denotes the boundary between the area of inner SFPs (x_max = 1), and the area of outer SFPs (x_max > 1)

Figure 12 shows that, for single particles with SFPs inside the beam core, the condition $k_{\text{p}\mathrm{,}\text{min}}=\sqrt{1-{\mu }_x}$ is always satisfied. For SFPs outside the beam core (${\widehat{x}}_{\text{max}}>1$), we obtain $k_{\text{p}\mathrm{,}\text{min}}>\sqrt{1-{\mu }_x}$. For a sufficiently large $\widehat{\delta }$, k_p,min > k_b/2. In this case, the 2:1 resonance cannot be driven by the breathing mode. In other words, the large off-momentum deviations of a single particle can dampen the 2:1 resonance. To show this more clearly, the motions of single particles with $\widehat{\delta }=3.0$, affected by the breathing, quadruple, and dispersion modes are shown in Fig. 13(a)–(c), respectively. The 2:1 resonances driven by the breathing and quadrupole modes are dampened because of the large momentum deviation $\widehat{\delta }$, and only the 1:1 resonance driven by the dispersion mode can be observed. Panel (d) shows the mixed case. Only the 1:1 resonance caused by the dispersion mode exists. The locking of the wavenumber is presented in panel (e), where the condition $k=k_{\text{d}}$ is satisfied, indicating that the 1:1 resonance is a parametric resonance.

Fig. 13Full-screen View

(Color online) Poincaré sections of single particles with $\widehat{\delta }=3.0$ affected by the breathing mode (a), quadrupole mode (b), dispersion mode, (c) and mixed modes (d), respectively, with Λ = 1.0 and μx =0.1. The wavenumbers (normalized to the dispersion mode) of single particles for mixed modes as a function of initial positions are shown in (e); the maximum displacement of single particle as a function of initial displacement is shown in (f), compared with the simulation results (red dotted lines)

PIC simulations were conducted using the parameters listed for Λ=1.0, as shown in Table 2, and the simulation results are shown in Fig. 13(f). The simulation results of the maximum displacements agreed closely with the numerical calculations based on the dispersion-modified PCM.