Wannier threshold law for two electron escape in the pres- ence of an external electric field

– We consider double ionization of atoms or ions by electron impact in the presence of a static electric field. As in Wanniers analysis of the analogous situation without external field the dynamics near threshold is dominated by a saddle. With a field the saddle lies in a subspace of symmetrically escaping electrons. Near threshold the classical cross section scales with excess energy E like σ ∼ Eα, where the exponent α can be determined from the stability of the saddle and does not depend on the field strength. For example, if the remaining ion has charge Z = 2, the exponent is 1.292, significantly different from the 1.056 without the field. The threshold behavior of double electron escape from an atom or ion was first tackled by Wannier in a famous paper in 1953 [1, 2, 3]. On the basis of the classical dynamics of two electrons in an atom he concluded that on account of the electron repulsion the two escaping electrons are correlated and that the cross section increases with excess energy E like σ(E) ∝ E, with a non-integer exponent α that depends on the charge of the remaining ion. Since the original paper many fine details of the process have been elucidated in both theory and experiment (for recent reviews see [4, 5, 6]). Electron–electron correlations are also important in the time-dependent process of double ionization in strong laser pulses. Measurements of the ion and electron momenta show that in non-sequential double ionization the escaping electrons prefer symmetry related motions [7, 8, 9]. The processes that are important for this double ionization are a matter of debate, but when discussed within the rescattering model [10, 11] similarities to the Wannier problem show up [12, 13, 14]. In the rescattering model one electron is temporarily ionized by tunneling, but driven back to the atom when the field changes the phase. During this rescattering event a highly excited two electron complex close to the nucleus is formed which then decays towards double ionization. For the field strengths where the characteristics of correlated electron escape are observed it seems crucial that the external field does not vanish when the decay takes place; otherwise the electrons do not have enough energy for double ionization [12, 13]. Typeset using EURO-TEX 2 EUROPHYSICS LETTERS In our previous presentations of the process in a time-dependent field we argued that the electron motion is fast compared to the field oscillations so that an adiabatic analysis can be applied. But it is possible to discuss the process also in the presence of a static field, where no adiabatic assumption is needed, and to push the analogy to the Wannier problem further by deriving the threshold laws for non-sequential double ionization in the presence of a static field. This is our aim here. Wanniers analysis divides into two parts: the identification of the configuration that leads to double ionization at the threshold and the determination of the threshold law from the stability exponents of the fixed point. In Wanniers case the threshold configuration had two electrons escaping symmetrically on opposite sides of the nucleus, thus minimizing electron repulsion. The presence of the field introduces a preference for motion in the direction of the field gradient. At the threshold for the process the energy is equally distributed between the two electrons. Furthermore, the distance of the electrons to the nucleus should be the same for otherwise any difference would be amplified by electron repulsion and the electrons would not escape simultaneously. Therefore, in the presence of an electric field the configuration that corresponds to Wanniers is one where the electrons escape along trajectories which are reflection symmetric with respect to the field axis [12, 13].