Electron-Electron Correlations and Dipole Selection Rules in Double Ionization of
Helium by Proton Impact
M. Schulz1,3, D. Fischer2,R. Moshammer2, and J. Ullrich2
1
University of Missouri-Rolla, Physics Department and Laboratory for Atomic,
Molecular, and Optical Research, Rolla, Missouri 65409, USA
2
Max-Planck Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany
3
Universität Frankfurt, Institut für Kernphysik, August-Euler Str. 6, D-60486
Frankfurt/Main, Germany
Dipole selection rules and electron-electron correlation effects in double
ionization of He by 6 MeV proton impact were studied by analyzing the
correlation function differential in the polar emission angles of both
electrons. In the data features are observed with a level of detail which is
comparable to fully differential cross sections.
Therefore, this method
opens unexpected possibilities to study effects due to dipole selection rules
and electron-electron correlations in unprecedented detail.
Studies of electron-electron correlation effects have been among the hot topics in
Atomic Physics for several decades [e.g. 1-5].
Such effects present an interesting
challenge to theory because they lead to an atomic potential which for each electron does
not only depend on its own coordinates, but also on the ones of the other electron. As a
result, the potential cannot be treated as an effective central or, in the case of an atomic
collision, two-centre potential. Our understanding of electron-electron correlations is still
rather incomplete, especially their dynamics in e.g. collision processes.
From a theoretical point of view, perhaps the simplest process for which dynamic
correlations can be studied is double ionization of helium by single photon impact [e.g. 610]. Here, the atomic potential of the target atom is not affected by the projectile and the
transition induced by the photon can to very good approximation be treated as an electric
dipole transition of one electron with the second electron being ejected due to electronelectron correlations.
Experimentally, fully differential cross sections have been
measured for a broad range of photon energies [11] and are usually presented as a
function of the emission angles of both electrons. The features observed in these twodimensional spectra are amazingly simple and can basically be understood by the dipole
selection rules in combination with the Coulomb repulsion between the electrons in the
continuum [12]. The dipole selection rules, evaluated by Maulbetsch et al. [13], state that
the sum momentum vector of both electrons Pe cannot be perpendicular to the direction
of polarization and the relative angle 12 = 1 - 2 between two electrons with equal
energy originating from a 1s2 state cannot be 180o.
For 2 keV electron impact Dorn et al. [14] found for selected kinematic settings
similar features as for photon impact. They analyzed double ionization events where
2
both electrons are emitted into the scattering plane defined by the initial projectile
momentum po and the momentum transfer q (difference between initial and scattered
projectile momenta). By selecting small momentum transfers and equal electron energies
favorable conditions for the dipole approximation were realized. Nevertheless, apart
from these “photon-like” features, some signatures of higher-order effects were also
observed. For proton impact under otherwise identical kinematic conditions the data
reveal even more striking similarities to photo- double ionization in that here higher-order
contributions are hardly visible at all [15].
In spite of these similarities between photon and charged particle impact in the gross
features, the details of the cross sections for the latter contain much richer information
about the reaction dynamics than for the former. In the case of photon impact the actual
transition (described by the dipole matrix element) is basically understood and the
accuracy of calculations is mostly limited by the details of the wavefunctions in the initial
and final states. For charged particle impact, in contrast, the transition itself is much
more complex involving a larger variety of different reaction mechanisms, which in a
perturbative expansion are described by different transition matrix elements.
More
specifically, dipole transitions are not as dominant as in the case of photo – double
ionization. Furthermore, higher-order contributions, which may or may not involve the
electron-electron interaction, are not always negligible even at small η. In fact at large η
they may be so strong that the expansion series does not even converge. Conceptually,
this problem is avoided in non-perturbative approaches which are currently, however,
nearly impossible to implement for double ionization by charged particle impact.
Moreover, this complicated dynamics of the two-center potential generated by the
3
projectile and the target nucleus (in the following referred to as the two-center potential)
has to be treated simultaneously with electron-electron correlation effects and these
contributions are very difficult to disentangle.
Recently, we introduced a new method of analyzing electron-electron correlation
effects which appears to be basically independent of the dynamics of the two-center
potential [16,17]. It is based on the correlation function, which is given by the ratio of
any two-electron spectrum (e.g. the momentum difference spectrum between the two
ionized electrons) for two electrons ionized in the same collision to the corresponding
spectrum for two independent electrons ionized in two different collisions. Here, we
analyze the correlation function for two-dimensional spectra as a function of the emission
angles of both electrons.
The experiment was performed at the 12 MV tandem Van de Graaf accelerator at the
Max-Planck Institut für Kernphysik in Heidelberg. A pulsed 6 MeV proton beam with a
pulse length of about 1 nsec and a repetition rate of 680 kHz intersected with a cold
(<1K) He beam from a supersonic gas jet. The recoil ions and the ionized electrons were
extracted in the longitudinal direction (defined by the initial projectile direction) by a
weak electric field of 2.3 V/cm. Their momentum vectors and the recoil charge state
were determined by using position sensitive detectors and time of flight techniques,
where a fast signal from the projectile beam pulser served as a timing reference. A
uniform magnetic field of 14 G confined the transverse motion of the electrons so that all
electrons with a transverse momentum of less than 1.5 a.u. were guided onto the detector.
For each double ionization event both electrons were detected simultaneously with a
single detector employing a multihit technique (dead-time  10 nsec). From the electron
4
momenta it is straightforward to calculate the emission angles. In the data analysis losses
due to the limited acceptance were carefully accounted for (for detail on acceptance
angles, dead-time effects and resolutions see [15,16]).
In Fig. 1a the fully differential cross sections of Fischer et al. [15] integrated over all
momentum transfers and all electron energies are plotted as a function of the polar
electron emission angles 1 and 2. Here, both electrons are ejected into the scattering
plane within an azimuthal angular acceptance angle of  =  10o. A polar angle of 0o
corresponds to the forward direction, 90o to the direction of the transverse component of
q, and 270o to the direction of the transverse component of –q. Since these cross sections
depend on four quantities (the polar and azimuthal angles of both electrons) we refer to
them as 4-fold differential cross sections 4DCS. Pronounced maxima are observed in the
two areas where the electrons were emitted at angles of about 15o and 120o. Between
these maxima, there is a minimum stretching along the entire diagonal for which the
difference between both angles is 0. This minimum is partly due to the experimental
limitation that electrons moving in the same direction with a small energy difference
won’t be identified as two separate particles in the detector because of the multihitdeadtime (indicated as dashed line in the figure).
However, this diagonal is also
suppressed by the Coulomb repulsion between the electrons in the continuum. It should
be kept in mind that in contrast to the equal-electron-energy case analyzed in [15], here
the entire two-dimensional spectrum is potentially affected by dead-time and
experimental acceptance effects and this spectrum is therefore not suitable for a
quantitative analysis of the cross sections.
5
The solid lines in Fig. 1a (and all following plots) indicate the regions which in photo
double ionization for two electrons of equal energies are prohibited by the requirement
that the electrons cannot be emitted back to back. No indication for a suppression of the
intensity along these lines is found. This is not surprising since double ionization events
leading to equal electron energies constitute only a relatively small fraction of the 4DCS
and the corresponding dipole selection rule does not hold for asymmetric energy sharing
among the electrons. Two regions which are subject to the other dipole selection rule
mentioned above, the requirement that Pe cannot be perpendicular to q, are indicated by
the crosses in Fig. 1a. Since the data are integrated over all momentum transfers, we
determined the direction of the momentum transfer averaged over all recorded double
ionization events and found an approximate angle of 75o relative to the initial projectile
beam direction. For the crosses, the individual electron momenta are both perpendicular
to q (1 = 165o and 2 = -15o) and 12 is 180o. Therefore, Pe must be perpendicular to q
as well, regardless of electron energies. Although this is a stricter selection rule, in that it
is not limited to electrons of equal energy, while the cross sections are generally small in
this area, again no minimum structures near the crosses are found.
The discussion of the 4DCS above shows that it is rather difficult to identify the
dipole selection rules even in highly differential cross sections in the case of charged
particle impact. Although their signatures are quite significant in fully differential cross
sections, they fade relatively fast upon successive integration over various kinematic
quantities. Eventually, features due to the selection rules become completely buried in
the structures due to the complicated dynamics of the two-center potential mentioned
above.
6
We recently demonstrated that effects due the two-center potential are strongly
suppressed in the correlation function [16] which will therefore be evaluated in the
following. The original correlation function is given by
R(p) = (dσ/d(p))cor/(dσ/d(p))unc –1,
(1)
where (dσ/d(p))cor is the measured cross section for finding two electrons emitted in the
same multiple ionization event with a certain momentum difference p and (dσ/d(p))unc
is the corresponding cross section for two independent electrons emitted in two different
collision events. Here, we analyze the 4-fold differential correlation function
R(1,2) = 4DCS(1,2)cor/4DCS(1,2)unc – 1,
(2)
i.e. the 4DCS of Fig. 1a are normalized to the corresponding spectrum for two
independent electrons from two different collisions, where the dependencies on 1 and
2 and the plane defining azimuthal angles are combined in the solid angles 1 and 2.
The limited acceptance for detecting 2 electrons, which, as mentioned above, affects
4DCS(1,2)cor, apriori has no influence on 4DCS(1,2)unc because here the electrons
originate from different collisions. However, we corrected 4DCS(1,2)unc for these
effects by eliminating all events in which the difference in time of flight between each
electron relative to its respective projectile is within the multihit-deadtime of 10 ns. In
R(1,2) these effects therefore cancel each other in 4DCS(1,2)cor and
4DCS(1,2)unc.
R(1,2) is shown in Fig. 1b. Qualitatively different structures compared to the
absolute 4DCS are quite obvious. Most noticeably, R(1,2) is large in areas where the
4DCS are rather small (upper left and lower right corners). Furthermore, we observe
minima along a large part of the dipole selection rule lines. However, these minima are
7
interrupted at angle combinations around (-60o,120o) and (60o,240o). Parts of the minima
are easily explained by the dipole requirement that Pe cannot be perpendicular to q. For
fixed q, this should only occur at well-defined angular combinations (crosses in Fig. 1b),
however, it should be kept in mind that the direction of q (75o), relative to which this
selection rule is indicated by the crosses, only represents an average value. In reality, the
direction of q can be anywhere from 0 to 90o because the data are integrated over all q.
Therefore, all angular combinations ranging from (90o,-90o) to (180o,0o) with 12 = 180o
are potentially affected by this selection rule. On the other hand, this cannot explain a
suppression of double ionization in the range from (180o,0o) to about (220o,40o). These
parts of the minima are probably due to the dipole selection rule prohibiting back to back
emission of electrons with equal energy. The visibility of this selection rules gives
already a flavor of the sensitivity of R(1,2) because the data are integrated over all
electron energies.
The areas of large values of R(1,2) stretching along lines parallel to the minima
can now be explained by a combination of final state correlations and flux conservation.
Final state correlations, i.e. the Coulomb repulsion between the electrons in the
continuum, favor 12 = 180o.
However, the dipole selection rules counteract this
tendency, as discussed above. As a result, part of the flux is “pushed” to the areas
parallel to the line corresponding to 12 =180o leading to a reduced 12 of around 120o
(or equivalently 240o). Finally, the interruption of the minima at angle combinations
around (-60o,120o) and (60o,240o) need to be understood. As far as the latter angle
combinations are concerned, we notice that one electron is approximately emitted in the
direction of q. Since here 12  180o, Pe must be pointing along an axis parallel to q,
8
regardless of the electron energies.
Therefore, these regions may be viewed as
counterparts to those indicated by the crosses with reversed roles, i.e. the dipole selection
rules favor double ionization events leading to Pe pointing parallel to q. Similar features
were also observed in fully differential cross sections for electron impact [14]. Since this
effect is not restricted to electrons of equal energies, it outweighs the selection rule
suppressing back to back emission. It should be noted that in the same regions the
absolute 4DCS are very small. R(1,2) is nevertheless large because it enhances the
parts of the final-state wavefunction in which correlation effects are important, again
illustrating the high sensitivity of the correlation function.
The interruption of the minima near angle combinations of (-60o,120o) are perhaps the
most surprising observation. At first glance, there seem to be no connection to neither
the dipole selection rules or to the direction of q. However, a more detailed analysis
suggests that the behavior in these regions can be explained in terms of a combination of
dipole transitions favoring Pe pointing in the direction of q and the influence of the
projectile – target nucleus interaction. For the case of single ionization it was recently
shown that this interaction is surprisingly important even at very small η [18] and even
more important in double ionization [19]. More specifically, an elastic scattering of the
projectile by the target nucleus following or preceding the ejection of the electrons, can
lead to a rotation of the total momentum transfer q about the initial projectile beam axis
relative to the momentum qe transferred to the electrons only. If this rotation is between
0o and 90o, then the polar angles 1 and 2 remain unaffected, but if the rotation is
between 90o and 180o these angles will be mirrored on the projectile beam axis, i.e. 1
and 2 will be turned into -1 and -2 (or 360 - 1 and 360 - 2, respectively).
9
Therefore, a possible explanation for the behavior of R(1,2) at (-60o,120o) is basically
the same as for (60o,240o) if the projectile – target nucleus is accounted for.
Finally we note that the features discussed above for the 4-fold differential correlation
function even remain after further integration over all electron emission planes (i.e. over
all azimuthal emission angles of both electrons). Due to this integration the electron
momentum vectors are projected onto the scattering plane and 1and 2 are now the
angles of these projections relative to po (see Fig. 2).
The corresponding doubly
differential correlation function
R(1,2) = DDCS(1,2)cor/DDCS(1,2)unc – 1,
(3)
is shown in Fig. 3, where DDCS refers to doubly differential cross section. The features
in R(1,2) are essentially the same as those discussed above for R(1,2). Therefore,
even after 6-fold integration (two electron energies, two azimuthal electron angles, the
azimuthal projectile angle, and the magnitude of the momentum transfer), the structures
due to the selection rules in the correlation function are as pronounced or even more
pronounced than in the absolute fully differential cross sections.
In summary, we have applied the correlation function to doubly and 4-fold
differential cross sections as a function of the polar emission angle of both electrons
ejected in double ionization. We have demonstrated that R(1,2) and R(1,2) very
sensitively reveal effects due to dipole selection rules and electron-electron correlations.
Structures which are not visible at all in the absolute cross sections are quite pronounced
even in R(1,2). A dipole selection rule which only holds for electrons with equal
energies is identified although the data are integrated over all energies. Even effects due
to the projectile – target nucleus interaction seem to be visible in R(1,2) if they occur
10
in combination with dipole selection rules or electron-electron correlations. Most of the
features discussed here are also observable in fully differential cross sections (FDCS), but
considering that the present data correspond to a 6-fold integral of the FDCS over various
kinematic quantities, the rich information it contains is remarkable. More importantly,
applying the correlation function to more differential cross sections, possibly even FDCS,
has the promising perspective to study dipole selection rules and electron-electron
correlation effects in unprecedented detail.
One of us (M.S.) is grateful for the hospitality of the University of Frankfurt and the
Max-Planck-Institut für Kernphysik in Heidelberg. This work was supported by the
Mercator Program of the Deutsche Forschungsgemeinschaft and the National Science
Foundation under grant Nos. PHY- 0353532 and INT-0224943.
Figure Captions
Fig. 1: 4-fold differential double ionization cross sections as a function of the polar
emission angles of both electrons for 6 MeV p + He collisions are shown on the left. The
solid lines indicate the dipole selection rule prohibiting back to back emission of
electrons with equal energy.
The crosses show the points where the sum electron
momentum is perpendicular to the average momentum transfer (see text) regardless of
electron energies. The right part shows the 4-fold correlation function R(1,2) (see
text), where 4DCScor is the 4-fold differential cross section shown on the left.
Fig. 2: Sketch of the electron emission geometry to illustrate the definition of the polar
angles 1 and 2 as used in the doubly differential data (see text). The thin solid arrows
11
are the projections of the electron momenta pe1 and pe2 onto the scattering plane (defined
by q and po) and 1 and 2 are the angles between these projections and po.
Fig. 2: Doubly differential orrelation function R(1,2) (see text). The sold lines and
crosses have the same meaning as in Fig. 1.
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240
240
180
180
x
120
350.0 -- 400.0
300.0 -- 350.0
250.0 -- 300.0
200.0 -- 250.0
150.0 -- 200.0
100.0 -- 150.0
50.00 -- 100.0
0 -- 50.00
x
2
2
120
60
60
0
0
x
-60
x
-60
-60
0
60
120
180
240
-60
0
1
60
120
180
240
1
Fig. 1a
Fig. 1b
pe2
2
p0
pf
1
q
pe1
Fig. 2
14
240
180
x
2
120
60
0
x
-60
-60
0
60
120
180
240
1
Fig. 3
15