Evidence of triple collision dynamics in partial photo-ionisation cross sections of helium

Experimental results on partial photo-ionisation cross sections of helium are analysed in the light of recent advances in the semiclassical theory of two-electron atoms. Byun et al [1] predict that the total photo-ionisation cross section below the double-ionisation threshold can, semiclassically, be described in terms of contributions associated with classical orbits starting and ending in the triple collision. The necessary modifications of the semiclassical theory for partial cross sections is developed here. It is argued that partial cross sections are also dominated by the triple collision dynamics. The expected semiclassical contributions can be identified in the Fourier transformation of the experimental data. This clearly demonstrates for the first time the validity of the basic assumptions made in [1]. Our findings explain furthermore in a natural way the self-similar structures observed in cross section signals for different channel numbers. PACS numbers: 32.80.Fb,03.65.Sq,05.45.Mt,05.45.-a Evidence of triple collision dynamics in partial photo-ionisation cross sections of helium 2 The rich resonance spectrum of two-electron atoms below the double ionisation threshold has been explored both experimentally and numerically up to principle quantum numbers N ≈ 13 − 17 of the remaining one-electron atom after ionisation [2, 3, 4, 5, 6, 7, 8, 9, 10]. Progress towards even higher N values thus moving closer to the three-particle break-up threshold E = 0 are hampered experimentally by the limited photon energy resolution and numerically by the high-dimensionality of the system. The complexity of the classical dynamics and the large number of degrees of freedom of the system have so far also restricted semiclassical calculations of individual resonances to subsets of the full spectrum and again small N values [11, 12]. Until recently it was thought that similar restrictions also apply to a semiclassical treatment of total and partial photo-ionisation cross sections for E < 0. The resonance density increases dramatically for energies approaching the double-ionisation threshold and individual resonances overlap and interfere leading to a strongly fluctuating cross section signal which decreases in amplitude towards the threshold [4, 5, 6]. The strong interaction between resonance poles can be regarded as a signature of the underlying chaotic classical dynamics which is also reflected in the resonance spacing distribution which shows a gradual transition towards that of random matrix theory approaching the double ionisation threshold from below [2, 6, 10]. We will consider in the following partial photo-ionisation cross sections of twoelectron atoms in the asymptotic regime E → 0−. Near the threshold, electronelectron correlation effects dominate which can be observed directly in scaling laws such as Wannier’s celebrated threshold law for double ionisation [13, 14] or in slow electron ionisation experiments taken across the threshold [15]; a cusp-like structure in this ”zero-kinetic energy” ionisation cross section is observed which has been interpreted in terms of classical escape along the Wannier ridge leading again to a threshold law with Wannier’s exponent both below and above E = 0 [16]. Zero-kinetic energy spectroscopy below the double ionisation threshold does, however, not resolve the complex resonance structure of the three-particle compound and the experimental signal contains thus little information about the mostly chaotic scattering dynamics of the underlying classical three-body Coulomb system. Semiclassically, total photo-ionisation cross sections can be described in terms of closed orbit theory (COT). The theory was developed for systems such as hydrogen in external fields [17, 18] or in the context of quantum defect theory for many electron atoms [19, 20, 21] for which the dynamics near the origin is regular. Recently, the necessary modifications for a closed orbit treatment of the total photo-ionisation cross section for two-electron atoms have been presented in [1]. Starting point of a COT is the total cross section in dipole approximation written in the form σ(E) = −4π α ~ωI〈Dφi|G(E)|Dφi〉 (1) where φi is the wave function of the initial bound state, D = π·r is the dipole operator, π is the polarisation of the incoming photon with angular frequency ω, and G(E) is the Green function of the system at energy E = Ei+~ω; furthermore α = e /~c is the fine-structure constant. In the semiclassical limit, the support of the wave function φi shrinks to zero relative to the size of the system reducing the integration in (1) to an evaluation of the Green function at the origin in the limit E → 0−. Writing the Green function in semiclassical approximation [17, 18, 22] thus leads to a summation over contributions from classical trajectories starting and ending at the origin. For two-electron atoms, the origin r = (r1, r2) = 0 represents the point where both electrons reach the nucleus simultaneously, that is, all three particles collide. Note Evidence of triple collision dynamics in partial photo-ionisation cross sections of helium 3 that we work in the infinite nucleus mass approximation, that is, the position of the nucleus is fixed at the origin. The presence of such triple collisions demands a careful re-evaluation of COT in the light of the three-body dynamics near the origin. The triple-collision itself forms a non-regularisable singularity of the classical equations of motion, that is, trajectories ending in a triple collisions can not be continued through the singularity. Trajectories coming close to a triple collision become extremely sensitive to initial conditions and nearby orbits approaching the collision point can be scattered into arbitrarily large angles. The triple collision singularity is in that sense infinitely unstable. This is in contrast to binary collisions which can be regularised by a suitable space and time transformation such as described in [23] leading to a smooth phase space flow in the vicinity of the collision. A semiclassical treatment of photo-ionisation starting from (1) needs to take into account classical trajectories beginning and ending near the three-body collision R = 0 where R = (r1 + r 2 2) 1/2 is the hyper-radius; the set of closed orbits usually employed in COT, namely those emerging out of and returning exactly to the triple collision point R = 0, are infinitely unstable and give a vanishing contribution to semiclassical expressions. These closed triple collision orbits (CTCO) act, however, as guiding centres for phase space regions leaving and returning to the triple-collision region. Taking the semiclassical limit E → 0− is equivalent to R0 → 0 in appropriately rescaled coordinates where R0 characterises the size of the initial wave function φi. By considering these limits carefully, it has been shown in [1] that the amplitude of the fluctuations in the total photo-ionisation cross section decays with a power-law according to σfl ∝ |E|μ with predicted exponent