Scattering off an oscillating target: Basic mechanisms and their impact on cross sections

We investigate classical scattering off a harmonically oscillating target in two spatial dimensions. The shape of the scatterer is assumed to have a boundary which is locally convex at any point and does not support the presence of any periodic orbits in the corresponding dynamics. As a simple example we consider the scattering of a beam of noninteracting particles off a circular hard scatterer. The performed analysis is focused on experimentally accessible quantities, characterizing the system, like the differential cross sections in the outgoing angle and velocity. Despite the absence of periodic orbits and their manifolds in the dynamics, we show that the cross sections acquire rich and multiple structure when the velocity of the particles in the beam becomes of the same order of magnitude as the maximum velocity of the oscillating target. The underlying dynamical pattern is uniquely determined by the phase of the first collision between the beam particles and the scatterer and possesses a universal profile, dictated by the manifolds of the parabolic orbits, which can be understood both qualitatively as well as quantitatively in terms of scattering off a hard wall. We discuss also the inverse problem concerning the possibility to extract properties of the oscillating target from the differential cross sections.

Figure 1

An arbitrary scatterer with convex surface is drawn in two positions: When its center is at the coordinate origin (dashed line) and at a displaced position where an impact takes place (solid line). Indicated is also a beam of particles as well as the important vectors that specify the motion of the scatterer and a projectile before and after the collision event.

Figure 2

3D plots for the vertical wall of the Gedankenexperiment: (a) The total number of collisions $\ell$ as a function of $v_0$ and ${\xi }_1$, (b) the outgoing velocity $v_{\mathrm{out}}$ as a function of $v_0$ and ${\xi }_1$. Very similar results hold for a single $b$ zone of the inclined wall or the disk.

Figure 3

(a) The escape velocity $v_{\mathrm{out}}$ as a function of ${\xi }_1$ for $v_0=1$ and the vertical wall. (b) The scattering function $v_{\mathrm{out}}$ versus the impact parameter $b$ for a single $b$ zone for the oscillating inclined wall. (c) The same as in (b) but for the oscillating disk.

Figure 4

1D model of the scattering off an oscillating vertical wall. The sinusoidal curve displays the position of the scatterer as a function of time $t$. The straight line segments represent the movement of different particles which are ejected asynchronously. The LV orbits [corresponding to ${\epsilon }_{1,\mu }$ (dashed line) and ${\epsilon }_{1,M}$ (dashed-dotted line) near ${\xi }_1=0$] which define the border between 1CEs and 2CEs are presented. They both approach tangentially the sinusoidal curve at the second point (near phase $3\pi ∕2$). The orbit labeled ${\epsilon }_1^{*}$ (solid line) leads to the maximum outgoing velocity for 2CEs.

Figure 5

Characteristic quantities for the scattering off the oscillating inclined wall ($\gamma =\pi ∕3$, $v_0=1$): The scattering functions (a) $v_{\mathrm{out}}\left(b\right)$, (b) $\theta \left(b\right)$, and the PDFs (c) ${\varrho }_{v_{\mathrm{out}}}$, (d) ${\varrho }_{\theta }$.

Figure 6

Characteristic quantities for the scattering off the oscillating disk ($\alpha =0$, $v_0=1$): The scattering functions (a) $v_{\mathrm{out}}\left(b\right)$, (b) $\theta \left(b\right)$, and the PDFs (c) ${\varrho }_{v_{\mathrm{out}}}$, (d) ${\varrho }_{\theta }$.

Figure 7

The values of successive extrema of $v_{\mathrm{out}}$ corresponding to $b^{*}$ obtained utilizing the exact model (◻) and the SDA (○).

Figure 8

Characteristic quantities for the scattering off the disk with oscillation angle $\alpha =\pi ∕3$: The scattering functions (a) $v_{\mathrm{out}}\left(b\right)$, (b) $\theta \left(b\right)$, and the PDFs (c) ${\varrho }_{v_{\mathrm{out}}}$, (d) ${\varrho }_{\theta }$.