Nuclear-electron versus nuclear-nuclear interactions in
ionization of atoms by highly charged ions impact
J. Fiol∗ and R. E. Olson∗
∗
Physics Department, University of Missouri-Rolla, Rolla, MO 65401, USA
Abstract. The correlated behavior of the electron and recoil-nucleus ion is studied for atomic ionization by collisions with
fast, highly charged ions. Calculations were performed using classical-trajectory Monte Carlo and quantum-mechanical
Continuum Distorted Wave methods that incorporate all interactions including the internuclear potential. Electron double and
triple differential ionization cross sections are presented as a function of projectile momentum transfer for hydrogen ionization
by 3.6 MeV/u Au53+ impact. It is possible to identify collision mechanisms that give rise to binary–like peak electrons, and
a new class of electrons recently predicted for highly charged ions impact, which were termed “swing by electrons”. The
relative importance of each mechanism is determined by the range of the nuclear-nuclear versus that of the nuclear-electron
interactions. An experimentally accessible method for the separation of these different mechanisms based on the study of the
azimuthal angle of the particles is proposed.
INTRODUCTION
THEORY
Differential cross sections as a function of the momentum transferred in the collision gives valuable information about the collision dynamics. While the ionized
electron momentum distribution by itself is nearly independent of the nuclear-nuclear interaction, description
of cross sections related to the projectile scattering requires a precise inclusion of all the interactions on the
same footing [1, 2]. Both the classical and quantummechanical methods employed in this work meet this
condition.
Differential cross sections as a function of the projectile momentum transfer for hydrogen and helium ionization by 3.6 MeV/u Au53+ became the focus of recent
theoretical and experimental works [3, 4, 5]. However,
serious discrepancies have been observed between theory and experiment for He ionization. Investigation of the
correlation between different fragments could provide a
better insight into the dynamics of the process.
The aim of this work is to discuss some of the threebody correlations in the ionization of ground and excited
states of hydrogen. Although the experimental production of atomic hydrogen is difficult, the present results
are closely related to cross sections of ionization of alkali atoms, which have only one active electron. Experimental study of ground and laser-excited states of these
atoms show a promising future by means of magnetooptical traps [6, 7, 8].
Both the Classical Trajectory Monte Carlo (CTMC) and
quantum mechanical theories employed in this work
have been described previously (see [5] and references
therein). CTMC hydrogen ionization cross sections were
computed with a modified three-body model that employs a Wigner distribution for the description of the
initial state [9]. This method yields a good description
of both momentum and coordinate distributions of the
atomic bound states.
The quantum-mechanical calculation of the differential cross sections relevant to this work are carried on
through (atomic units are used)
Z
dσ
dσ
=
dΩ
dQ⊥ dEe
dQ⊥ dEe dΩe
dσ
(2π )5
=
k Q⊥ |ti f |2
dQ⊥ dEe dΩe
v2
(1)
, ti f = hΨ−
f |Vf |Ψi i
where Q⊥ is the component of the momentum transfer
perpendicular to the initial projectile velocity v. The
parallel component of the momentum transfer is fixed by
energy conservation Qk ≈ (Ee + |εi |) /v, with Ee and εi
the final and initial electron energy, respectively. Here ti f
is the transition matrix element between the initial and
final states. The final state employed is the C3 or CDW
wavefunction,
~
Ψ±
f (r, R) =
~
~
ei (kT ·~rT +KT ·RT ) ± ± ±
DN DP DT
(2π )3
(2)
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
138
-14
where the distortion factor Dα is defined in terms of the
wavefunction
10
53+
3.6 MeV/u Au
+ H(n)
~
eikα ·~rα ±
D ,
(2π )3/2 α
(3)
n=1
dσ/dEe dQ⊥ (cm /a. u. )
ψ~kα (~rα ) =
Nν
10
n=2
2
2
solution of the scattering two-body problem [10]. In the
case of Coulomb interactions Dα is given by [11]
h
i
Dα± = Nν 1 F1 ∓iνα ; 1; −i(kα rα ∓~kα ·~rα ) (4)
-15
= Γ(1 ± iνα ) e−π να /2
where να = mα Zα /kα is the Sommerfeld parameter for
the interaction Zα /rα between a given pair of particles,
~kα is their relative momentum, and mα is the correspond~ α are the position and
ing two-body reduced mass. ~Rα , K
the momentum of the remaining particle relative to the
pair. The perturbation potential Vf in the evaluation of
the transition matrix ti f with the wavefunction (2) is
-16
10
n=3
-17
10
0.1
1
2
3 4 5
Q⊥ (a. u.)
∇D−∗
α
, (5)
D−∗
α
FIGURE 1. CDW-B1 (lines) and CTMC (symbols) double
06. J Fiol and R E Olson.
differential crossFigure
section
of ionization of 50 eV electrons from
"Three-body dynamics in hydrogen ionization by fast ..."
ground and excited states of hydrogen (from Ref. [5]).
where MP and MT are the masses of the projectile and
target-nucleus, respectively ([5] and references therein).
Approximations employing (2) and (5) are referred as
Continuum distorted wave (CDW) theories (CDW-B1
when an initial Born state is used).
momentum transferred by the projectile to a target particle in a two-body classical collision is
Vf =
KP KN KT KN
−
− KT KP
MP
MT
,
Kj =
Qα⊥ =
2ν α
,
bα
(6)
where bα is the impact parameter between the two particles and να is the corresponding Sommerfeld’s parameter. The momentum transferred
to an electron emitted
p
with 50 eV is Qe = k = 2(Ee + |εi |) ≈ 2.16 a. u. While
the longitudinal component of the momentum transfer is
Qk = (Ee + |εi |)/v ≈ 0.19 a. u., the projectile-electron
impact parameter is be ≈ 4.1 a. u. For H(1s) the characteristic electron-nucleus distance is rat ≈ 1.5 a. u. Then,
the projectile-nucleus impact parameter ranges 2.6 ≤
bR ≤ 5.6 a. u., with momentum transfer 1.5 ≤ QR⊥ ≤
3.5 a. u. The momentum transferred to the nucleus points
opposite to that transferred to the electron, so that they
tend to cancel each other in vector addition, resulting in
the small overall momentum transfer Q⊥ observed.
In the case of excited states, the characteristic size of
the target atom is larger than for ground state (rat ≈ 5
and 11 for n = 2 and 3, respectively). The interaction
between the projectile and the target-nucleus is weaker,
giving rise to the binary-like peaks observed in the spectra of figure 1. Binary-peak structures are characteristic
of perturbative collisions where the range of the interaction is smaller than the size of the atom. In fact, this kind
of structures have recently been reported for ionization of
helium and hydrogen by impact of C6+ [3, 4, 5, 12]. For
electron energy emission of 50 eV the binary peak is lo-
RESULTS AND ANALYSIS OF THE
DYNAMICS
Figure 1 shows double differential cross sections
(DDCS), defined by (1), for ionization of ground (n = 1)
and excited (n = 2, 3) states of hydrogen by impact
of 3.6 MeV/u Au53+ . Comparison between the results
obtained within the Classical Trajectory Monte Carlo
(CTMC) and quantum-mechanical Continuum Distorted
Wave - Born Initial state (CDW-B1) agree very well
with each other for ionization of 50 eV electrons. Both
theories predict similar characteristics in the cross section as a function of the perpendicular component of the
momentum transferred in the collision. For ground-state
hydrogen ionization, the major contribution comes from
low momentum transfer collisions. On the other hand,
ionization spectra of excited n = 2 and n = 3 states show
a pronounced peak at relatively high momentum transfer.
As previously observed, these characteristics are due
to the larger atomic radial distributions for increasing
values of n [5].
In order to further investigate the origin of the differences in the DDCS lets consider the effect of the projectile over each of the target fragments. The perpendicular
139
cated at momentum transfer Q⊥ ≈ 2.1 a. u., as it has been
observed for C6+ . However, the spectra of figure 1 for
ionization of H(n) present the maxima shifted to lower
values (Q⊥ ≈ 1 and 1.5 for n = 2 and n = 3 respectively).
Observe that those shifts are in agreement with the analysis in terms of relative distances between the three particles. While the distance between projectile and electron
remain unchanged from the previous analysis the impact
parameter projectile-nucleus is bR ≤ 9.1 a. u. for n = 2.
The maximum momentum transferred to the nucleus is
QR⊥ ≈ 1 a. u., producing the shift observed in figure 1. A
similar analysis for state n = 3 shows that the momentum transferred to the recoil-ion is QR⊥ ≈ 0.5 a. u., also
coinciding with the observed shift.
tron is larger than that transferred to the recoil-ion. The
curves for Q⊥ = 1.6 a. u. show a peak at larger angles corresponding to the so-called “swing-by” electrons.
They were attributed to processes with close projectilenucleus collisions [5]. However, currently there is not
available experimental data to confirm or refute such prediction.
A convenient representation of the correlation between
the participating particles is given by the cross section
as a function of the azimuthal angle between the different fragments. Figure 3 presents the CTMC differential ionization cross section as a function of the azimuthal angle of the projectile relative to the recoil ϕPR
(horizontal axis) and to the electron ϕPe (vertical axis).
Observe that binary collisions projectile-electron would
180
Let’s return our attention to the ionization of the 1s state
of hydrogen. In this case the perpendicular momentum
transferred by the projectile to the nucleus-target ranges
from 1.5 a. u. to 3.5 a. u. Thus, the momentum transferred
to the nucleus can be larger or smaller than the transferred to the electron. In a previous paper it was predicted
that these two different processes lead to different signatures in kinematically-complete cross sections [5]. Figure 2 shows the triple differential cross sections (TDCS)
for electron emission of 50 eV electrons in the scattering
plane. For small momentum transfer the peak is located
150
ϕPe (degrees)
ELECTRON-RECOIL CORRELATION
120
90
60
30 CTMC
53+
Au
0
0
53+
+ H(1s) → Au
30
60
+
-
+H +e
90
120
150
180
ϕPR (degrees)
40
Q⊥=0.8 a. u.
30
-17
dσ/dEedΩedΩP (10
FIGURE 3. Azimuthal angle of the projectile relative to the
recoil-ion and electron. The scale is linear with darker shades
corresponding to larger values of the cross section.
Q⊥=1.6 a. u. (x 20)
2
cm /a. u.)
35
25
contribute only to the region of ϕPe = 180◦ . Similarly,
pure projectile-recoil collisions are confined to the plane
ϕPR = 180◦ . For these collisions the maximum of the
cross section occurs in the diagonal region, for ϕRe =
180◦ , similarly to the observed in photoionization processes. However, in this case it does not correspond to
photon-like collision mechanisms but is rather due to
the high charge of the projectile. While the projectilenucleus distance bR is larger than the electron-nucleus
distance rat , the projectile pushes the two target fragments in opposite directions.
In figure 4 are presented similar CTMC results than
those of fig. 3, but only with events where the final momentum of the recoil-ion KR is either larger or smaller
than the electron momentum ke . We observe that the
maximum observed in figure 3 is composed of two separable contributions. The upper figure, displaying only
events when the final electron momentum is larger than
20
15
10
5
0
0
60
120
180
240
300
360
Electron polar angle (degrees)
FIGURE 2. CDW-B1 (lines) and CTMC (symbols) TDCS of
H(1s) ionization by impact of 3.6 MeV/u. The final momenta
of the three particles are on the same plane. The electron energy
is 50 eV.
near the position of the binary projectile-electron collision (marked with an arrow). This case corresponds to
collisions where the momentum transferred to the elec-
140
CONCLUSIONS
180
The three-body dynamics in ground and excited hydrogen ionization by fast, highly charged ions has been investigated. Two collision mechanisms are identified. One
of them, producing binary-like spectra, corresponds to
a stronger projectile-electron interaction. On the other
hand, the named “swing by” electrons are produced
when the projectile suffers a closer collision with the
target-nucleus than with the electron. However, in both
cases the projectile interacts with the two target-particles
simultaneously and a two-body approximation is not
valid. An experimentally practical method for separate
these processes has been proposed, which is based in the
study of the azimuthal angle of the three particles.
150
120
90
ke > KR
ϕPe (degrees)
60
30
0
150
ACKNOWLEDGMENTS
120
Support from the Office of Fusion Energy Sciences
(DOE) is gratefully acknowledged
90
KR > ke
60
REFERENCES
30
0
Rodríguez, V. D., J. Phys. B: At. Mol. Phys., 29, 275–286
(1996).
2. Fiol, J., Rodríguez, V. D., and Barrachina, R. O., J. Phys.
B: At. Mol. Phys., 34, 933–944 (2001).
3. Olson, R. E., and Fiol, J., J. Phys. B: At. Mol. Phys., 34,
L625–L631 (2001).
4. Moshammer, R., Perumal, A., Schulz, M., Rodríguez,
V. D., Kollmus, H., Mann, R., Hagmann, S., and Ullrich,
J., Phys. Rev. Lett., 87, 223201 (2002).
5. Fiol, J., and Olson, R. E., J. Phys. B: At. Mol. Phys., 35,
1759–1773 (2002).
6. van der Poel, M., Nielsen, C. V., Gearba, M. A., and
Andersen, N., Phys. Rev. Lett., 87, 123201 (2001).
7. Tursktra, J. W., Hoekstra, R., Knoop, S., Meyer, D.,
Morgenstern, R., and Olson, R. E., Phys. Rev. Lett., 87,
123202 (2001).
8. Flechard, X., Nguyen, H., Wells, E., Ben-Itzhak, I., and
DePaola, B. D., Phys. Rev. Lett., 87, 123203 (2001).
9. Hardie, D. J. W., and Olson, R. E., J. Phys. B: At. Mol.
Phys., 16, 1983–1996 (1983).
10. Fiol, J., and Olson, R. E., Accepted for publication in
Nucl. Instr. and Meth. B (2002).
11. Brauner, M., Briggs, J. S., and Klar, H., J. Phys. B: At.
Mol. Phys., 22, 2265–2287 (1989).
12. Schulz, M., Moshammer, R., Madison, D. H., Olson,
R. E., Marchalant, P., Whelan, C. T., Walters, H. R. J.,
Jones, S., Foster, M., Kollmus, H., Cassimi, A., and
Ullrich, J., J. Phys. B: At. Mol. Phys., 34, L305–L311
(2001).
1.
0
30
60
90
120
150
180
ϕPR (degrees)
FIGURE 4. Azimuthal angle distributions as in figure 3. See
text for details.
the momentum of the recoil-ion (ke > KR ), contributes
to the region ϕPR < 90◦ and ϕPe > 90◦ . The lower plot in
figure 4 exhibits opposite features. The main contribution
occurs for ϕPR > 90◦ and ϕPe < 90◦ .
The cross sections observed in figure 4 are consistent
with an interpretation in terms of relative distances between the particles. First, observe that, as stated before,
the electron and recoil-ion are bound to be emitted in
opposite directions because the electron is attracted by
the projectile while the nucleus is repelled. In the first
case, ke > KR , the projectile has a closer collision with
the electron than with the recoil-ion. The projectile and
the electron are scattered in approximately opposite directions in a binary-like collision while the recoil-ion,
which suffers a milder collision, is emitted in the opposite direction. In the lower plot are selected only those
events where the projectile is mainly scattered by the
target-nucleus (KR > ke ), indicating that the projectile approaches closer to the recoil-ion than to the electron.
141