Observing interference effect in binary (e, 2e) of molecules

We present investigations on the two-center and multi-center interference effects of molecules in binary (e, 2e) experiments. The high energy resolution electron momentum spectroscopy (EMS) measurements on H2 are reported with final vibrational states resolved. The experimental momentum profiles for ionization transitions to the individual final vibrational states of the ion are obtained. The measured and calculated vibrational ratios of the cross sections deviate from Franck-Condon principle, which can be ascribed to the Young-type two-center interference. Furthermore, with the help of our latest version of EMS spectrometer which has considerably higher sensitivity and much wider momentum range from 0 to 8 a.u., we are able to extend our observations to multi-center interference effect in high symmetry molecules like NF3 and CF4 with several oscillation periods included.


Introduction
As we all know, particle-wave duality of matter particles plays key role in quantum mechanics. It is one of the prominent conceptual deviations from the classical physics. This revolutionary concept has been directly demonstrated by the beautiful electron double-slit experiment by Clause Jönsson in 1961 [1]. Since then, double-slit experiments have shown the wave character of increasingly larger quantum objects, including fullerenes (buckyballs) [2] and huge organic molecules [3]. Another way to realize the double-slit experiment is the coherent superposition of electrons emitted from two indistinguishable atoms in diatomic molecules [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. This is often referred to as molecular double-slit interference, which was first suggested by Cohen and Fano [4] in 1966 in photoionization of diatomic molecules N2 and O2. Such interference effects will lead to the energy-or angle-dependent oscillations in cross sections. It took 35 years before this two-center interference effect was unambiguously proven for H2 in the ionizations by heavy ions [5], and another 4 more years in 2005 to find evidence in photoionization of N2 [10].
The interference effect in dipole (e, 2e) ionization of H2 was first predicted by theoretical calculations in 2003 [17] and then observed by experiments employing coplanar asymmetric kinematics at intermediate energies [18,19]. The effect was revealed from the suppression or enhancement of the forward (binary) or backward (recoil) scattering peaks as compared to helium at same kinematics. In fact, it would be more straightforward to observe Young-type interference in binary (e, 2e). For binary (e, 2e), we have (1) high electron impact energy; (2) larger momentum transfer to ejected electron. If the Bethe ridge condition is satisfied that the incident electron transfers all the lost momentum to target electron, the triple differential cross section (TDCS) of (e, 2e) [23] is then directly linked to the square modulus of the single-electron wave function in momentum space and can be used to directly 'imaging' the electron momentum distributions for individual orbitals. The relevant technique is usually terminized as electron momentum spectroscopy (EMS).
For the simplest diatomic molecule H2, as a simple approximation, the molecular orbital (MO) ψ can be expressed by a linear combination of atomic orbitals (LCAO): where 1sA and 1sB are two identical 1s atomic orbitals centered on atoms A and B of the molecule. It is easy to obtain the TDCS [17] (3) where (3) H σ is the atomic TDCS, R0 is the internuclear distance at equilibrium and q is the magnitude of the recoil momentum of residual ion which is equal to p, the magnitude of the momentum of orbital electron, under the EMS condition. In principle, we can directly observe the interference factor 0 0 1 sin( ) / ( ) pR pR + by plotting the ratio of TDCSs of molecule and atom as a function of momentum p. Such phenomena is also called bond oscillation [24,25] and its prediction can be traced back to the early 1940s [26]. However, for H2, it is difficult to observe such interference effect for two reasons. Firstly, the momentum range for a conventional EMS instrument is limited to 0 ~ 3.0 a.u. A much wider range of momenta up to 2π/R0 = 4.5 a.u.is at least required to observe a complete period of bond oscillation in H2, having the internuclear distance R0 of 1.4 a.u. Secondly, the EMS cross section decreases very rapidly as momentum increases. In order to have clear observations of interference effect, one way is to choose a molecule having larger internuclear distance. Most recently, the multi-centre interference effect was observed in binary (e, 2e) experiment for the three outermost MOs of CF4 [22], each of which are consisted of a combination of non-bonding 2p AOs located on the four F atoms. The F-F internuclear distance is 4.02 a.u. and the period 2π/R0 = 1.6 a.u. is well located in the momentum range of EMS. Another way to observe the interference effect is to compare the interference factor at different internuclear distance. In this talk, we present our first measurement [27] on vibrationally resolved EMS of H2 by a high-resolution (e, 2e) spectrometer [28]. The experimental momentum profiles for ionization transitions to the individual final vibrational states of the ion are obtained. By choosing different vibrational states, we equivalently change the internuclear distance. The measured and calculated vibrational ratios of cross sections reveal obvious deviations from Franck-Condon values. Such deviations can be ascribed to the Young-type two-center interference. Furthermore, with the help of our latest version of EMS spectrometers [29] which have considerably higher sensitivity and much wider momentum range from 0 to 8 a.u., we are able to extend our observations to multi-center interference effect in high symmetry molecules like NF3 and CF4.

Background
EMS is based on the (e, 2e) experiment in which an electron from target atom or molecule is cleanly knocked out by a high-energy incident electron and the residual ion acts as a spectator. From energy and momentum conservation, the binding energy εf and the momentum p of the target electron are given by (4) where Ei, pi (i = 0, a, b) are kinetic energies and momenta of the incident and two outgoing electrons, respectively.
Within the binary-encounter approximation, as well as the plane wave impulse approximation (PWIA), the TDCS for (e, 2e) ionization is [23] Here fee is the electron-electron collision factor which is essentially constant in EMS conditions. Therefore the cross section is proportional to a structure term which is the square of overlap between the initial target state G and final ion state I. av Σ denotes a sum for final states and average for initial states that are not resolved in the experiment and are considered as degenerate. For molecular target the initial state G and final state I can be described in terms of Born-Oppenheimer approximation which is a product of separate electronic, vibrational, and rotational functions [23]: where Vµ and Dν are the vibrational and rotational functions for the initial state. The indices µ and ν represent quantum numbers that specify the vibrational and rotational states respectively. Final vibrational and rotational quantities are denoted by primes. The notations 0 and i represent the electronic states of target and ion. At room temperature the target is in its vibrational ground state 0 For the conventional EMS spectrometer, the energy resolution (typically 1~2 eV) cannot resolve the final vibrational and rotational states. The cross section reduces to For the H2 molecule Dey et al. [30] showed that the vibrational average gives the same results as taking the electronic functions at their equilibrium nuclear geometry. This approximation has subsequently been justified by agreement with a wide range of experiments. So the cross section further reduces to The integral in equation (9) is known as the spherically averaged electron momentum distribution, or electron momentum profile. For H2 molecule which has only one vibrational mode, the structure amplitude

Two-center interference effects in vibrationally resolved (e, 2e) of H2
where ( ) ( ) are again area-normalized to unity respectively for the convenience of comparison. Using Franck-Condon (FC) approximation, equation (11) is reduced to ( ) ) obviously deviates from constant (here is unity due to the normalization), especially the low quantum number group 'L' whose ratio declines sharply with the momentum. The deviation from the FC principle can be ascribed to the two-center interference effect. When taking into account vibrational states, the vibrationally resolved cross section can be approximated by where 0 σ is the equivalent one center atomic cross section. By replacing in equation (13) the variable R by characteristic value R µ′ associated with µ′ state, the vibrational ratio can further be approximated by This formula clearly predicts that the vibrational ratio should oscillate around the quotient of FC factor. To evaluate the observations, the turning points on potential curve for relevant vibrational states are adopted as the characteristic value R µ′ and the function respectively. The parameter a1 is introduced to compensate the approximations in the evaluations of R µ′ and thus is kept the same value for all three fittings. The fitted curves are presented in figure 3(a)-(c) as chain lines. The agreement of the model fitting with the measured and calculated vibrational ratios undoubtedly signifies the Young's two-center interference effect and the movement of the interference fringe has also been observed. Vibrationally resolved experiment provides a more straightforward way to observe Young-type interference in electron impact ionization of diatomic molecules, which does not rely on the comparison with one-center atomic cross section.

Multi-center interference effects in (e, 2e) of CF4 and NF3
As we have mentioned above, in order to have clear observations of interference effect, one way is to choose a molecule having larger internuclear distance. The multi-centre interference effect was observed in binary (e, 2e) experiment for the three outermost MOs of CF4 [22] recently. The large F-F internuclear distance makes a complete period of oscillation to be observable. Most recently, a high-sensitivity angle and energy dispersive multichannel electron momentum spectrometer with simultaneous detection in 2π angle range has been developed [29]. The sensitivity of EMS has been improved by employing a double half wedge and strip anode (DH-WSA) position-sensitive detector (PSD) combined with a 90° sector, 2π spherical electrostatic analyzer. Furthermore, much wider momentum range from 0 to 8 a.u. has been achieved, which makes it possible to include more than two periods of oscillation in observations of multi-center interference effect in high symmetry molecules like NF3 and CF4. Here, only the results of CF4 are presented.
inner valence outer valence The three outermost orbitals are non-bonding, essentially due to the 2p lone-pair electrons on fluorine atoms. For these MOs consisting of the F 2p AOs, the TDCS can be expressed as [22] ( ) where RFF is the internuclear distance between the F atoms and C0 and C2 are coefficients of the spherical Bessel functions of order 0 and 2 respectively. The EMS cross section σ2p(p) for the 2p AO of an isolated F atom is calculated by distorted wave Born approximation employing B3LYP/aug-cc-pVTZ wavefunction. The function [1 + C0j0(pRFF) + C2j2(pRFF)] governs the oscillatory structure and hence it is the interference factor in this case  The interference factors for the three outermost orbitals of CF4 are shown in figure 4. It is immediately clear that the experiments exhibit oscillatory structures. Compared to the previous experiment [22], more periods of oscillations are included, further confirming the interference effect. To highlight it more closely, the function h[1 + C0j0(pRFF) + C2j2(pRFF)] is subsequently employed as a fitting curve for 4t2 to reproduce the experiment with RFF, h, C0, and C2 being fitting parameters. The best fit to the experiment is presented also in figure 4(b) by the dashed line. The resulting RFF value is 3.9 Bohr which is in excellent agreement with 4.07 Bohr reported by electron diffraction [32].