Dynamic two-center resonant photoionization in slow atomic collisions

An additional channel opens for photoionization of atom $A$ by an electromagnetic field if it traverses a gas of atoms $B$ resonantly coupled to this field. We show that this channel, in which $A$ is ionized via resonant photoexcitation of $B$ with subsequent energy transfer to $A$ through two-center electron correlations and which is very efficient when $A$ and $B$ constitute a bound system, can strongly dominate the ionization of $A$ also in collisions where the average distance between $A$ and $B$ exceeds the typical size of a bound system by orders of magnitude.

We shall assume the collision to be slow enough such that practically no excitation (or ionization) of the colliding partners is possible if A and B enter the collision being in their ground states. This is the case if w  a v 1 fi 0 (Massey adiabatic criterion, see e.g. [24]), where ω fi and a 0 are typical transition frequency and linear size, respectively, of A and/or B and v is the collision velocity. (We note that the condition w  a v 1 fi 0 roughly corresponds to impact energies below 1 keV u −1 .) However, if atom B is coupled to an electromagnetic (EM) field resonant to a dipole transition between its ground and excited states, then the incident atom A can be ionized by absorbing the excitation energy of B via dynamic two-center electron correlations.
We shall consider only very distant collisions, in which the interaction between A and B is quite weak and their nuclei move practically undeflected. In distant collisions electron transfer processes are not very likely and can be neglected. Indeed, in such collisions the overlap of electronic states of A and B is per se always very small. Moreover, due to the relative motion of the atoms this overlap effectively diminishes further since even in closer collisions there might be not enough time for an electron to make a transfer to the other atomic center.
Even though the collision velocity is low, we shall assume that the relative motion of the nuclei can still be regarded as classical (the corresponding conditions are discussed, for instance, on pp 108-111 of [25], they are fullfilled down to impact energies as low as ∼1 eV u −1 ).
In a reference frame, where atom B is at rest and taken as the origin, atom A moves along a classical straightline trajectory R(t)=b+vt, where b=(b x , b y , 0) is the impact parameter and v=(0, 0, v) the collision velocity. In this frame the collision is described by the equation where the total Hamiltonian is given by is the Hamiltonian of a free (non-interacting) atom A (B), and is the interaction between A and B, where r (x) is the coordinate of the electron of A (B) with respect to the nucleus of A (B). The 'electrostatic' approximation (3) for the inter-atomic interaction can be used if the distance R is not too large: w  R c tr , where c is the speed of light and ω tr the frequency of the virtual photon transmitting the interaction (see e.g. [18,22,26]).
with the external EM field which will be taken as a classical linearly polarized field F=F 0 cos(ωt−k·(R+r )) ( , where F 0 is the field strength, ω the field frequency and k the wave vector (F 0 ·k=0). The interactionsŴ A andŴ B read ) with A 0 =−cF 0 /ω is the vector potential of the EM field at the position of the electron of atom A (B) andp r ( x p ) is the momentum operator for the electron of atom A (B). Below these interactions are taken in the dipole approximation: k · r=0, , we also set k·R=0. We first include the interaction between atom B and the EM field by replacing the ground state f 0 (with an energy ò 0 ) and the excited state f 1 (with an energy ò 1 ) of non-interacting atom B by its field-dressed bound states where a  ( ) t 0 and a  ( ) t 1 are time-dependent coefficients to be determined. We assume that the field is switched on adiabatically at  -¥ t and impose the boundary conditions where Δ=ò 0 +ω−ò 1 is the detuning, G B rad the width of the excited state f 1 due to its spontaneous radiative decay and = - Using the states (5), the first order perturbation theory with respect to the interactionV AB , and keeping in mind that at  -¥ t both atoms were in the ground states we obtain that the two-center ionization amplitude for atom A reads where ψ 0 with an energy ε 0 is the ground state of atom A, ψ p with an energy ε p describes an electron emitted with an asymptotic momentum p, and f ± are determined by equations (5)- (7). We note that all these quantities refer to the rest frame of A. Besides, in our derivation we have neglected the Doppler shift and the coupling to the scalar potential which appears in a moving reference frame [27] that is justified due to the low collision velocity.
Performing the integration over time in (8) we obtain We note that Δ p /v represents the minimum momentum transfer in the collision. Further, and K n (n=0, 1, 2) are the modified Bessel functions [28].
The differential cross section, which describes the spectra of electrons emitted via the two-center channel in collisions with impact parameters where the integrations run over the azimuthal angle j b of b and its absolute value. In particular, if the field is polarized along the z-axis, we obtain after rather lengthy calculations that is the radial matrix element for transitions between the ground and continuum states of atom A with u 0 and u p,1 being their radial parts (the ground state of atom A was assumed to be an s-state and u p,1 denotes the continuum radial wave with the orbital quantum number l=1).
where d 0 and d 1 are the radial parts of the ground and excited states of atom B.
The functions K n (x) (n=0, 1, ..) diverge at x→0 and decrease exponentially at x>1 [28]. Therefore, in distant low-velocity collisions ( the main contribution to the total cross section stems from a very small interval of emission energies centered at ε p,r =ε 0 +ω with width δε p ∼v/b. Since δε p is much less than a typical energy range Δε p in which the quantity r p ). Then the contribution σ 2c to the total cross section from collisions with  b b min is given by 2 . The mean free path for the EM field in the gas of atoms B is given by λ=1/(n B σ excit ) and it has to be larger than the size of the gas target.
Having derived the two-center ionization rate we shall now briefly discuss the single-center photoioization.
Single-center ionization. The amplitude for the direct (single-center) ionization of atom A is given by In stronger fields, where rad ) and the first order of perturbation theory in the interactionŴ B is no longer valid, the so called rotating-wave approximation can be used instead [30]. One can show that in such fields the ratio η becomes smaller and decreases with increasing the field.
Let us now apply equation (19) to three collision systems. 0 the mean free path λ of the radiation in a gas of silicon atoms is about 1.6 cm. Thus, for this collision system a substantial enhancement of PI from distant collisions due to the two-center channel would be possible for gas targets not exceeding ∼2 cm. Unlike the rates  c 1 and  c 2 their ratio η does not depend on the transition matrix element of A. Therefore, we can apply (18), (19) also if 'atom' A is in fact a molecule. In particular, for photo-dissociation of I 2 ( e » | | 1.57 0 eV) in collisions with Li (the 2s-2p transition, ω≈1.85 eV) we obtain an enhancement which is almost as strong as for the H − -Rb system.
Since, according to our estimates, the inclusion of the contribution from collisions with b<b min (not taken into account here) strongly increases η, the effectiveness of the two-center channel may be regarded as spectacular: in a gas of atoms B with n B ∼10 10 cm −3 the average distance between the atoms A and B is about 2.5×10 −4 cm∼10 4 -10 5 a.u. and nevertheless the two-center mechanism may still strongly dominate ionization of A.
We note in this context that within the present approach the magnitude of b min cannot be strictly defined. As was just mentioned, closer collisions (with b<b min ) are expected to yield a very substantial contribution to twocenter PI. Therefore, in an attempt to account for as much of the total rate as possible, in the above three examples the value of b min was chosen to be close to the minimum possible value of the impact parameter which still enables one to fulfill the main assumptions of our approach: the nuclei of the colliding particles move along straight-line trajectories, the electrons of the colliding atomic particles essentially do not overlap and the interaction between them can be treated in the dipole-dipole approximation within the first order of perturbation theory.
Due to a steep dependence of the two-center channel on the inter-atomic distance the colliding atoms interact mainly in the vicinity of their closest rapprochement (R∼b) that strongly reduces the effective distance. At a fixed interatomic distance R the two-center-to-single-center ionization ratio η 0 (at the resonance) reads h w  ( ) c R 0 6 [22]. Comparing it with equation (19) we see that the structure of both ratios is similar and that for ionization in atomic collisions the fixed interatomic distance R is replaced by the quantity is the average distance between the atoms. In this sense R eff can be regarded as the 'effective' distance between the atoms in the process of collisional two-center PI.
If the density of atoms B is not very high ( n b 1 1 0 B min 3 21 -10 22 cm −3 ) the 'effective' distance R eff turns out to be much less than the average distanceR. This explains why the two-center photoionization channel can still be so effective in collisions, where the average distance between A and B is enormous on the scale of a typical size of bound systems. Nevertheless, one should note that on this scale R eff is quite large as well. For instance, taking b min =10 a.u. and assuming that n B ;10 10 cm −3 we obtain R eff ; 10 3 a.u., which is much smaller than  R 10 5 a.u. but still much larger than a typical size of bound atomic systems. This shows that the effectiveness of the two-center photoionization channel in collisions will be overally much weaker than in bound systems.
Compared to two-center PI in the 'static' case [22], this process in collisions attains new features. In particular, both angular and energy spectra of the emitted electrons change qualitatively. For instance, the emission cross section (11) vanishes at an energy ε p =ε 0 +ω 0 , i.e. exactly where its 'static' counterpart reaches maximum [22]. This is caused by (destructive) interference between the contributions to the transition amplitude due to the incoming (t<0) and outgoing (t>0) parts of the trajectory R(t).
The present study focuses on slow collisions (10 eV u −1 E col 1 keV u −1 ) for which the connection with two-center reactions in bound systems is relatively close. In addition, using an approach in which all multipoles of the interatomic interaction as well as all impact parameters (b min =0) are taken into account, we considered two-center PI in energetic (10 keV u −1 ) collisions where the competing channels of direct impact ionization by the collision partner become effective. It turns out that two-center PI-due to comparatively very small momentum transfers involved in this process-can strongly outperform the 'normal' collisional ionization channels up to impact energies ;25 keV u −1 and is very 'visible' at higher energies as well. Besides, since the total cross section for two-center PI scales roughly ∼1/v up to ;25 keV u −1 , the ratio η of two-center and direct PI remains a constant up to such rather large impact energies.
For collisions considered in this study the retardation effects have a minor impact on the cross section but with increasing the transition frequency they will eventually become of importance. However, as our analysis shows, the standard theoretical approaches being applied to the retarded interaction in collisions, where the minimum momentum transfer is zero, yield unphysical results and thus suitable treatments still have to be developed.
In conclusion, photoionization of an atom A in an external EM field can strongly increase if it traverses a gas of atoms B which are in a dipole resonance with this field. This enhancement is caused by the transmission of photo-excitation energy from atom B to atom A via dynamic two-center electron correlations.
Two-center correlations are already known as an extremely efficient mechanism of 'communication' between parts of a bound system whose size R is typically of the order of few or several Bohr radii. Although in collisions the average distance between the atoms reaches tens of thousands of Bohr radii, the two-center correlations still turn out to be quite effective. This unexpected result forms an interesting connection between the area of interatomic phenomena and the field of atomic collisions: a large variety of interatomic processes extensively investigated in bound systems (e.g. various types of ICD [33], electron capture [34] and recombination [18,19], resonance scattering [19]) may play a role in collisions as well.
A Rb gas target with a diameter of ;1.4 mm and n B ∼10 10 cm −3 driven by three weak resonant (ω≈1.59 eV) continuous lasers currently functions at the Institute of Modern Physics. The lasers produce homogeneous (within a diameter of 14 mm) beams with intensity in the crossing area of ;1 W cm −2 transferring up to 20% of the atoms from the ground 5s 1/2 to the excited 5p 3/2 -state. It is planned to combine the target with a beam of ∼100 eV H − (current ;10 pA, beam size ;1 mm). Our estimates show that with the above parameters one can accumulate sufficient numbers of detachment events (with and without the Rb target) for measuring the predicted effects.