Post-Collision Interaction (PCI)

• Post-Collision Interaction (PCI) is defined as a family of post-event dynamics, encompassing Coulomb-mediated energy exchange in atomic physics and controlled recovery phases in robotics.
• In atomic and condensed-phase studies, PCI involves three-body interactions that lead to energetic chirps and asymmetric spectral line shapes, influenced by factors like screening and scattering.
• In fields such as robotics and heliophysics, PCI describes post-impact control and momentum exchange processes that enhance system recovery and improve arrival-time predictions.

Searching arXiv for relevant PCI papers across atomic physics, condensed phase, robotics, and heliophysics. Post-collision interaction (PCI) denotes distinct post-event interaction regimes in several research literatures. In atomic physics it describes Coulomb-mediated energy exchange after inner-shell ionization and Auger decay, or, in a related usage, time-delayed post-collisional ionization following creation of a vacancy. In collision-resilient robotics it denotes the closed-loop phase that begins after impact detection and comprises deformation recovery and trajectory replanning. In heliophysics it denotes the propagation and structural evolution of coronal mass ejections (CMEs) after they come into dynamical contact and exchange momentum and energy. Across these usages, PCI names processes that occur after the initiating collision or ionization event and that subsequently reshape spectra, charge states, trajectories, or propagation dynamics (Bauch et al., 2012, Montanari et al., 2016, Lu et al., 2021, Mishra et al., 2014, Dupuy et al., 24 Jul 2025).

1. Terminology and domain-specific meanings

The term is not uniform across disciplines. In the cited literature, five recurring usages can be distinguished.

Usage of PCI	Operational meaning	Representative paper
Auger-decay PCI	Energy exchange when a faster Auger electron overtakes a slower photoelectron	(Bauch et al., 2012)
Field-assisted / liquid PCI	PCI modified by a streaking field, screening, or scattering	(Bauch et al., 2012, Dupuy et al., 24 Jul 2025)
Post-collisional ionization	Time-delayed emission of additional electrons after creation of an inner-shell or sub-valence vacancy	(Montanari et al., 2016)
Robotic PCI	Detection, recovery control, and replanning after physical impact	(Lu et al., 2021)
CME post-collision interaction	Post-contact propagation of interacting CMEs through the heliosphere	(Mishra et al., 2014)

In atomic and condensed-phase spectroscopy, PCI is tied to sequential electron emission and Coulomb correlation. In the robotic formulation, PCI is explicitly operationalized as a distinct control phase between nominal motion segments. In the CME literature, PCI denotes an extended post-collision evolution rather than a brief impact instant. A recurrent source of confusion is that the same abbreviation labels mechanisms with very different state spaces and observables; the commonality is temporal and dynamical, not mechanistic.

2. Coulombic PCI in Auger decay

In the theoretical description of Auger decay, PCI is a three-body Coulomb interaction effect involving the photoelectron, the Auger electron, and the ion whose charge changes from singly to doubly charged during the process. The sequence is: XUV photoionization creates a core hole and a photoelectron; after a random delay $\tau_A$ governed by the Auger decay rate $\Gamma_A$, the hole decays and emits a faster Auger electron; if $E_A > E_P$, the Auger electron catches up with and overtakes the photoelectron at distance $r^*$ and time $t^*$. The abrupt change in the effective ionic potential at the overtaking point causes energy loss of the photoelectron and energy gain of the Auger electron, with the effect strongest when overtaking occurs close to the ion (Bauch et al., 2012).

The Auger delay is exponentially distributed,

$$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$

Because the overtaking distance grows roughly linearly with $\tau_A$, the PCI contribution decreases with delay, and in the simplified picture

$$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$

The paper further writes

$$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$

This maps decay time to final Auger energy and yields an energetic chirp: early Auger decays, occurring closer to the ion, undergo larger PCI shifts than late decays.

A central point is that this chirp is not a pulse-induced chirp. It arises from dynamical three-body Coulomb interaction rather than from any chirp of the XUV pump pulse. The same work also shows that coincidence spectra exhibit strong diagonal correlation, reflecting energy gained by the Auger electron and lost by the photoelectron, whereas the photoelectron spectrum does not exhibit a chirp and instead shows nearly time-independent PCI broadening.

3. Field-assisted and condensed-phase PCI

In field-assisted PCI, the Auger sequence unfolds in the presence of a streaking field with vector potential $A_L(t)$. Without PCI, the final momentum shift is approximately $\Gamma_A$0, and for the Auger electron the detector energy is shifted by the vector potential at emission time. Including PCI yields a combined time-to-energy mapping,

$\Gamma_A$1

so the observed Auger-energy shift is the sum of a linear or weakly quadratic streaking chirp and a hyperbolic PCI chirp. At zero crossings of $\Gamma_A$2, this produces a pronounced asymmetry: for $\Gamma_A$3, a broad, PCI-broadened line; for $\Gamma_A$4, a compressed line with a forbidden energy region below

$\Gamma_A$5

The asymmetry can be quantified by

$\Gamma_A$6

with $\Gamma_A$7 the FWHM for opposite slopes of the vector potential (Bauch et al., 2012).

The same post-overtaking idea persists in liquids, but two medium-specific modifiers become dominant: dielectric screening and electron scattering. In the liquid-phase study of solvated $\Gamma_A$8, $\Gamma_A$9, and $E_A > E_P$0, PCI is treated as a correlated two-electron effect involving a slow photoelectron and a faster KLL Auger electron emitted sequentially after inner-shell photoionization. In a simple classical picture, the exchanged energy scales as $E_A > E_P$1, where $E_A > E_P$2 is the crossing radius. The free-atom semiclassical treatment gives an asymmetric Auger lineshape,

$E_A > E_P$3

which enhances intensity on the high-kinetic-energy side of the Auger line (Dupuy et al., 24 Jul 2025).

In liquids, screening reduces the interaction through

$E_A > E_P$4

while scattering increases PCI by keeping the photoelectron near the ion for longer. The paper uses $E_A > E_P$5 for water and reports PCI shifts of $E_A > E_P$6 eV on KLL Auger lines when the photoelectron kinetic energy is only $E_A > E_P$7 eV above threshold. The three isoelectronic ions show very similar PCI behavior, and changing the solvent from water to methanol or ethanol has little effect. A key conclusion is that screening and scattering can nearly cancel: screening weakens the Coulomb energy exchange, whereas elastic and inelastic scattering increase the effective interaction time, returning the net PCI magnitude to values comparable to free atoms once condensed-phase broadening is accounted for. The work further reports that Monte-Carlo electron-transport simulations reproduce the measured shifts and asymmetric lineshapes more faithfully than a simple effective-damping modification of semiclassical PCI theory.

4. Post-collisional ionization and branching-ratio formalisms

A related but distinct meaning of PCI is “post-collisional ionization,” defined as time-delayed electron emission following creation of a single inner-shell or sub-valence-shell vacancy. The process begins with single ionization of subshell $E_A > E_P$8, producing $E_A > E_P$9, followed by Auger decay, autoionization, or Auger cascades that emit one or more additional electrons. If exactly $r^*$0 additional electrons are emitted during relaxation, the final ion is $r^*$1. In this literature PCI is treated as intrinsic to the target and independent of the projectile, because Auger lifetimes are typically much longer than the collision time (Montanari et al., 2016).

The central quantities are the branching ratios $r^*$2, where $r^*$3 is the initially ionized subshell and $r^*$4 is the number of post-collisionally emitted electrons. They obey the unitarity relation

$r^*$5

These tabulated probabilities are given for $r^*$6 to $r^*$7 for Ne, Ar, Kr, and Xe. Their physical interpretation is direct: $r^*$8 corresponds to no PCI electron and a final $r^*$9 ion; $t^*$0 corresponds to one PCI electron and a final $t^*$1 ion; and so on up to $t^*$2, which yields a final $t^*$3 ion.

The data show strong shell and atomic-number dependence. For Ne $t^*$4, $t^*$5, so one additional electron is the dominant outcome. For Ar $t^*$6, $t^*$7, so a $t^*$8 vacancy almost always generates two PCI electrons. For deep Kr and Xe shells, emission of $t^*$9, $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$0, or $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$1 additional electrons becomes common. By contrast, valence shells such as Ar $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$2 and $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$3, Kr $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$4 and $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$5, and Xe $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$6 and $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$7 have $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$8 and $$\mathcal{P}_{T_A}(\tau_A) = \Gamma_A e^{-\Gamma_A \tau_A}, \qquad \tau_A > 0.$$9, reflecting that Auger-type PCI is energetically forbidden there.

These branching ratios enter multiple-ionization modeling through projectile-dependent single-ionization cross sections and projectile-independent PCI probabilities. At the cross-section level the paper writes

$\tau_A$0

so the entire projectile dependence is contained in $\tau_A$1, while the post-collisional relaxation is encoded in $\tau_A$2. This formalism is used to model multiple ionization by electrons, positrons, protons, antiprotons, and other positive ions.

5. PCI in collision-resilient mobile robotics

In the robotic literature, PCI is not a spectral effect but a control architecture for impact-resilient systems. The process is divided into four phases: pre-impact nominal motion along a polynomial trajectory; impact detection and estimation of the immediate post-impact state; a post-impact recovery phase over a fixed horizon $\tau_A$3; and post-impact nominal motion generated from the recovered state. The paper explicitly states that “detection, deformation-phase control, and replanning” is what it operationalizes as post-collision interaction (Lu et al., 2021).

The robot is a holonomic planar mobile robot with a deflection ring connected to the main chassis by four passive visco-elastic prismatic joints. Hall effect sensors measure the deformation vector $\tau_A$4, and collision is detected when $\tau_A$5 exceeds a threshold. A collision frame $\tau_A$6 is instantiated at collision time $\tau_A$7, with basis $\tau_A$8, where $\tau_A$9 is normal to the deflection vector. During recovery, the state is

$$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$0

and the control input is

$$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$1

The compliant post-impact dynamics are modeled by a Voigt-type deformation equation,

$$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$2

supplemented by a collision-frame state-space model with normal elastic and damping terms in $$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$3 and tangential friction coupling in $$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$4. Recovery is posed as a constrained optimal-control problem over $$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$5,

$$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$6

subject to linearized dynamics, spring-limit constraints, and terminal conditions including $$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$7. The desired terminal velocity is chosen from the remaining nominal segment toward the next waypoint, clamped to avoid inward motion into the obstacle and scaled to respect a global maximum velocity $$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$8.

Once recovery ends, the post-impact state $$\Delta E^{\text{PCI}} \propto \frac{1}{r^*} \propto \frac{1}{\tau_A}.$$9 becomes the initial condition for a minimum-effort piecewise polynomial replanner with waypoint and continuity constraints. If the next waypoint is occluded in the collision frame, Algorithm 2 either adjusts it or inserts an intermediate exploration waypoint. Nominal planning imposes only waypoint constraints and does not perform continuous collision checking; unexpected contact simply triggers another PCI cycle.

The reported experiments use a $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$0 environment, a 12-camera VICON system, and an onboard Intel NUC with control at $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$1 Hz. Hall-based collision detection missed only $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$2 collisions. In a case where the original trajectory did not intersect the obstacle surface, the deformation-recovery-and-replanning strategy was $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$3 faster and $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$4 shorter than the collision-avoidance baseline, with slightly higher control energy and an end-point error increased by $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$5. In a case where the original trajectory intersected the obstacle surface, it was $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$6 slower and $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$7 longer, with higher control energy but an end-point error $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$8 smaller than the collision-avoidance strategy. The work therefore treats impact as a controlled, model-informed event rather than as planning failure.

6. PCI in interacting coronal mass ejections

In the CME literature, PCI denotes the post-collision or post-interaction phase after two CMEs have come into dynamical contact, exchanged momentum and energy, and then continued toward $$\Delta E^{\text{PCI}}(\tau_A) \approx \frac{1}{p_r \tau_A + \delta r^*}.$$9 AU as modified, interacting structures. In the November 9–10, 2012 event, the leading edge of the November 10 CME interacted with the trailing edge of the November 9 CME. The collision phase is defined operationally as the interval during which the slower structure accelerates and the faster one decelerates until their speeds become approximately equal or the acceleration trends reverse (Mishra et al., 2014).

The measured collision began around 2012 Nov 10, 11:30 UT and ended around Nov 10, 17:15 UT. Over that interval, the trailing edge of CME1 accelerated from $A_L(t)$0 km s$A_L(t)$1 to $A_L(t)$2 km s$A_L(t)$3, while the leading edge of CME2 decelerated from $A_L(t)$4 km s$A_L(t)$5 to $A_L(t)$6 km s$A_L(t)$7. Using true masses $A_L(t)$8 kg and $A_L(t)$9 kg, the paper infers a best-fit coefficient of restitution $\Gamma_A$00, consistent with a strongly inelastic collision and close to perfectly inelastic behavior. CME1 momentum increases by $\Gamma_A$01, CME2 momentum decreases by $\Gamma_A$02, and the total kinetic energy decreases by about $\Gamma_A$03.

After the collision phase, the two structures propagate with nearly equal speeds of about $\Gamma_A$04–$\Gamma_A$05 km s$\Gamma_A$06, remain dynamically linked, and produce a distinct interaction region between them. WIND observations identify a magnetic cloud associated with CME1, an interaction region with heating, compression, magnetic holes, elevated $\Gamma_A$07, and signatures interpreted as magnetic reconnection, and a separate flank encounter of CME2. The paper emphasizes that the CMEs do not fully merge into a single complex ejecta; instead, PCI produces a coupled system with recognizable substructures.

This post-collision evolution materially improves arrival-time prediction. When post-collision speeds at the last HI tracking point are used as input to a drag-based model, predicted arrival errors are much smaller than those obtained by simply propagating the COR2 three-dimensional speeds. The paper states that using post-collision kinematics reduces arrival-time errors by about half a day or more, especially for CME2. It also links the strongest geomagnetic response not to the shock alone but to the trailing edge of CME1 plus the interaction region, where sustained southward $\Gamma_A$08, enhanced density, and elevated ram pressure drove the main storm and intense substorms.

7. Misconceptions, validity limits, and open problems

Several misconceptions recur across the PCI literature. First, the abbreviation does not identify a single universal mechanism. Atomic PCI, post-collisional ionization, robotic PCI, and CME PCI share a post-event temporal structure but not a common microscopic model. Second, in Auger streaking the energetic chirp is explicitly distinguished from XUV-pulse chirp: it is produced by dynamical Coulomb interaction rather than by the pump pulse itself (Bauch et al., 2012). Third, in liquids screening does not imply weaker net PCI; the liquid-phase study argues that scattering can compensate or even overcompensate screening, so condensed-phase PCI can remain comparable to the free-atom case (Dupuy et al., 24 Jul 2025). Fourth, the post-collisional ionization tables are restricted to Auger-type PCI and do not include mechanisms such as interatomic Coulombic decay in dimers (Montanari et al., 2016). Fifth, the robotic approach is collision-inclusive rather than collision-free and therefore is not intended for environments where collisions are strictly forbidden (Lu et al., 2021). Sixth, CME PCI is not a point-like binary impact; the cited analysis itself notes uncertainties in exact collision timing, three-dimensional structure, and mass evolution beyond COR2 (Mishra et al., 2014).

The major validity limits are domain specific. In field-assisted Auger theory, the TDSE is one-dimensional, the analytical model assumes short XUV pulses relative to the streaking cycle and an overtaking regime with fast Auger electrons, and the classical and analytical treatments neglect quantum interference, spin, and exchange. The liquid-phase treatment identifies unresolved issues in the use of classical versus semiclassical PCI line-shape reconstruction and in the quality of low-energy electron-scattering cross sections for water. The branching-ratio formalism treats $\Gamma_A$09 as energy independent once the vacancy is created, although the paper notes that extremely near thresholds or resonant conditions may require more subtle dependence. The robotic formulation assumes planar motion, a single dominant contact arm, fixed contact during recovery, simple Coulomb-like friction, and local waypoint heuristics rather than global obstacle-aware optimization. The CME analysis uses a one-dimensional collision approximation for intrinsically three-dimensional magnetized structures.

These limitations define the main open problems stated in the papers. For atomic streaking PCI they include the influence of XUV pulse chirp and substructure, more refined quantum treatments including interference and spin, and systematic exploration of the regime where photoelectron energy exceeds Auger energy. For liquid PCI they include refinement of scattering parameters in water, better treatment of decoherence and nonlocal dielectric response, and more rigorous coupling of transport Monte Carlo to semiclassical PCI theory. For post-collisional ionization they include unresolved valence-shell contributions in Ne and the treatment of non-Auger channels. For robotic PCI they include extension to 3D robots, multi-contact scenarios, more robust nonlinear control or MPC, and integration with higher-level planners. For CME PCI they include better handling of three-dimensional propagation, mass accretion, and single-point in situ ambiguities. Taken together, these open questions show that PCI is best understood not as a settled term of art but as a family of post-event dynamical problems whose detailed interpretation remains strongly context dependent.