Classical scattering of charged particles confined on an inhomogeneous helix

We explore the effects arising due to the coupling of the center of mass and relative motion of two charged particles confined on an inhomogeneous helix with a locally modified radius. It is first proven that a separation of the center of mass and the relative motion is provided if and only if the confining manifold represents a homogeneous helix. In this case, bound states of repulsively Coulomb interacting particles occur. For an inhomogeneous helix, the coupling of the center of mass and relative motion induces an energy transfer between the collective and relative motion, leading to dissociation of initially bound states in a scattering process. Due to the time reversal symmetry, a binding of the particles out of the scattering continuum is thus equally possible. We identify the regimes of dissociation for different initial conditions and provide an analysis of the underlying phase space via Poincaré surfaces of section. Bound states inside the inhomogeneity as well as resonant states are identified.

Figure 1

(a) Helix with a local modification of the radius $R\left(u\right)$ and a pitch $h$, as given by Eq. (11). (b) Local modulation of the radius as a function of the parameter $u$.

Figure 2

Potential curve for the uniform helix with parameters $h=0.4\pi$ and $R=1$. We observe three potential wells located at $s=3.34,10.00$, and $16.75$ with minimum values $V=0.48,0.36$, and $0.26$, respectively.

Figure 3

Contour plot of the potential $V(S,s)$ for the inhomogeneous helix. The dashed lines represent the positions of the minima of the three wells for the homogeneous case. The effect of the local modification of the radius is evident for $\left|S\right|\lesssim 30$.

Figure 4

Euclidean distances of the particles for the minimum ($Ⓧ,①$) and the maximum ($Ⓧ,②$) potential configurations of the first potential well for three different regions of the inhomogeneous helix: (a) the uniform domain of the helix with pitch $h$ and radius $R=1$, (b) the left side of the inhomogeneous region where the radius increases by $H_1$ at the bottom and by $H_2$ at the top, whereas the pitch remains the same, (c) the central part of the inhomogeneous domain $S\approx 0$, which since the modulation of the radius is stationary can be approximately treated as a part of a uniform helix with the same pitch $h$, but double radius $R=2$.

Figure 5

Overview of final bound and unbound states for initially bound states started in the first potential well: (a) Final potential values $V$ [blue (gray) dots] and final relative energy values $E_s$ (black dots) for different initial c.m. kinetic energies $T_S$. The dashed red lines represent the boundary potential values (minimum of well, maximum of barrier) of the first potential well. (b) Final relative coordinate values $s$ (black dots) for different initial c.m. kinetic energies $T_S$. The dashed red lines represent the boundary values of those $s$ which lie within the first potential well.

Figure 6

Finally bound (solid black line) and dissociated (dashed blue line) trajectories near (a) the first transition point $T_{S_{c1}}\approx 3.83$, (b) the second transition point $T_{S_{c2}}\approx 38.86$. The trajectories in each case [(a), (b)] differ in their c.m. velocities only by $0.001$.

Figure 7

(a), (b) Same as in Fig. 5, but for particles starting in the second potential well. (c), (d) Same as in Fig. 6, but with (c) $T_{S_{c1}}\approx 1.15$, (d) $T_{S_{c2}}\approx 33.54$.

Figure 8

(a), (b) Same as in Fig. 5, but for particles starting in the third potential well. (c)–(f) Same as in Fig. 6, but with (c) $T_{S_{c1}}\approx 0.46$, (d) $T_{S_{c2}}\approx 3.76$, (e) $T_{S_{c3}}\approx 5.59$, (f) $T_{S_{c4}}\approx 70.35$. The small subfigures in (e) and (f) present the respective trajectories for large values of $S$, following them up to the point where the dissociative and bound trajectories separate from each other.

Figure 9

(a) States that after scattering remain bound [cyan (light gray) regions] and states that are led to dissociation [blue (dark gray) regions], (b) color encoded values of the relative difference between the final and the initial maximum kinetic energy of the relative motion $(T_{{\max }_f}^s-T_{{\max }_i}^s)/E_s$, (c) color encoded values of the difference between the final relative energy $E_{s_f}$ and the maximum value of the potential barrier of the first well $V_{\max }$ for different $s_{0L}$ ($x$ axis), $\frac{t}{T}$ ($y$ axis), and ${\log }_{10}{T}_S$ ($z$ axis).

Figure 10

(a) Trajectories with different c.m. kinetic energies $T_S$ for $s_{0L}\approx 1.56$ and $\frac{t}{T}=0.22$. The numerical values presented in the diagram correspond to the values of $T_S$ for the different kinds of trajectories. (b) Trajectories with different phases of relative oscillation $\frac{t}{T}$ for $T_S=8.88$ and $s_{0L}\approx 1.56$. The numerical values presented in the diagram correspond to the values of $\frac{t}{T}$ for the different kinds of trajectories. In both cases, the dissociative trajectories of type $A$ are shown with solid white lines, while those of type $B$ are represented with solid black lines. The finally bound trajectories are presented with dashed cyan lines. The vertical black dashed line indicates the position $S_{hL}$ at which the hump starts (by definition).

Figure 11

Trajectories for various phases of relative oscillation $\frac{t}{T}$ for $T_S=8.88$ and (a) $s_{0L}=s_{\min }=3.34$, (b) $s_{0L}=3.26$, (c) $s_{0L}=2.53$, (d) $s_{0L}=1.56$. Solid white lines stand for finally dissociated states of type $A$, solid black lines for type $B$, and dashed cyan lines for finally bound states.

Figure 12

(a) The rate of change of the kinetic c.m. energy ${\dot{T}}_S$ as a function of $S,s$ for $\dot{S}=1$. The solid black lines depict the position of the maxima of the three potential barriers, while the dashed brown lines indicate the position of the minima of the three potential wells in the uniform regime (see also Fig. 3). (b) The dwell time as a function of the initial c.m. velocity ${\dot{S}}_0$ for the particles starting at the minimum of the first potential well with zero relative velocity (blue line with circles). For small ${\dot{S}}_0$, the deviation from a motion with constant $\dot{S}={\dot{S}}_0$ dictated by the red dashed line is evident.

Figure 13

EPLs of the first potential well for representative total energies: (a) $E=0.28$, (b) $E=0.39$, (c) $E=0.476$, (d) $E=0.6$, (e) $E=0.744$, and (f) $E=0.76$.

Figure 14

Poincaré surfaces of section for representative total energies: (a) $E=0.28$, (b) $E=0.39$, (c) $E=0.476$, (d) $E=0.6$, (e) $E=0.744$, and (f) $E=0.76$. The inner region of (c) (blue dots) consists of chaotic trajectories.

Figure 15

Selected trajectories (a), (c) and respective PSOS (b), (d) for the energies: (a), (b) $E=0.39$, (c), (d) $E=0.4761$. The color of the trajectories corresponds to their position in the PSOS. In (a), (c) the EPLs are also depicted as in Fig. 13.

Figure 16

(a) Selected bound trajectories, (b) PSOS, and (c) escaping trajectories in the c.m. $S$ coordinate for the energy $E=0.6$.

Figure 17

(a) Selected bound trajectories, (b) PSOS, and (c) escaping trajectories both in the c.m. $S$ (bound states) and in the relative coordinate $s$ (free states) for energy $E=0.76$.

Figure 18

A resonant trajectory for $E=0.6$ and initial conditions $S=0,s=5.134,\dot{S}=0.549,\dot{s}=-0.387$: (a) plot of the trajectory on the potential landscape, (b) time evolution of the c.m. coordinate $S$ of the trajectory.