Download Three-body dynamics in hydrogen ionization by fast highly charged

INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 1759–1773
PII: S0953-4075(02)32735-4
Three-body dynamics in hydrogen ionization by fast
highly charged particles
J Fiol and R E Olson
Physics Department, University of Missouri-Rolla, Rolla, MO 65401, USA
E-mail: [email protected]
Received 8 January 2002, in final form 15 February 2002
Published 26 March 2002
Online at stacks.iop.org/JPhysB/35/1759
Abstract
Electron double and triple differential cross sections are calculated as a function
of projectile momentum transfer for ionization of ground and excited state
hydrogen by 3.6 MeV u−1 C6+ and Au53+ ions. These three-body Coulomb
systems are investigated using the classical-trajectory Monte Carlo and
continuum distorted wave methods that incorporate all interactions including
the nuclear–nuclear potential. The calculations allow one to distinguish cross
section features associated with three collision mechanisms. The first is due
to distance collisions that lead primarily to a binary interaction between the
projectile and electron. The second is a strong three-body interaction by the
projectile with both the electron and the target ion when the projectile passes
between the electron and its parent nucleus. These two mechanisms result
in binary peak electrons located near the momentum transfer vector Q with
active participation by the recoil ion along −Q in order to determine the overall
momentum transfer magnitude. The third mechanism, which is present only
for fast highly charged ion impact, yields electron spectra that here-to-fore have
not been observed. Rather than the electrons being scattered near the angle θ
associated with the momentum transfer vector Q, they are found at an angle
of 360◦ − θ . Such electrons are due to a close collision of the projectile with
the target nucleus with the electron being forced to swing by its parent. These
electrons are not so-called recoil electrons associated with the angle 180◦ + θ.
Moreover, in this case neither the recoil ion nor the electron spectra peak near Q.
Calculations are also presented for collisions with excited hydrogen in order
to assess the collision dynamics as a function of the radial dimensions of the
target atom.
1. Introduction
Single ionization of an atom induced by a collision with an ion provides a challenging field
for the study of the three-body Coulomb problem. For many years the analysis of atomic
0953-4075/02/071759+15$30.00
© 2002 IOP Publishing Ltd
Printed in the UK
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J Fiol and R E Olson
ionization produced by heavy ion impact was mainly conducted through the study of total cross
sections, and differential cross sections in terms of the angle and energy of the ejected electron.
Only recently has the development of experimental techniques permitted the coincidence
measurement of both the ionized electron and recoil ion momentum vectors. In particular,
momentum spectroscopy experiments now provide kinematically complete descriptions of
heavy-particle three-body ionization collisions (Cocke and Olson 1991, Ullrich et al 1997).
These data can be used to rigorously test different theoretical approaches.
In electron–atom collisions the study of kinematically complete ionization cross sections
has been carried out both experimentally and theoretically for more than two decades (see
Ehrhardt et al (1985), Lahmann-Bennani (1991) and references therein). Research in this
area has shown that for relatively high incident velocities the first Born approximation (FBA)
gives reliable results even for highly differential cross sections. However, electron impact
studies are limited because of the fixed, low charge of the projectile. In the traditional picture,
the strength of the collision is estimated through the perturbation or Sommerfeld parameter
νP = ZP /v, where ZP and v are the projectile’s charge and initial velocity, respectively. For
electron impact the magnitude of the perturbation strength νP cannot exceed unity for ground
state hydrogen or rare gas targets.
The use of heavy ions provides an additional degree of freedom due to the possibility
of varying the projectile charge. Increasing the projectile charge makes it possible to
exceed perturbation strengths of unity. The description of ionization collisions in such high
perturbation conditions is theoretically much more tenuous. This fact was recently illustrated
by detailed studies of He single ionization. Double differential cross sections (DDCSs) were
presented for the impact of 3.6 MeV u−1 Au53+ ions (νP = 4.42) as a function of the projectile
momentum transfer (Moshammer et al 2001, Olson and Fiol 2001). Poor agreement between
experimental data and theoretical predictions was observed.
The study of DDCSs as a function of the momentum transfer in a collision gives
valuable information on the dynamics of three interacting particles. While the ionized
electron momentum distribution by itself is nearly independent of the nuclear–nuclear
interaction, a precise inclusion of all interactions on the same basis is needed in order to
properly describe cross sections related to the momentum transfer or projectile scattering
angle. Previous research has shown that the inclusion of the internuclear interaction is
necessary to describe ionization cross sections related to the projectile scattering angle
for large values of this variable (Salin 1989, Rodrı́guez 1996). Moreover, it has recently
been shown that the cross section for small deflections of the projectile is strongly affected
by the nuclear–nuclear interaction in both classical and quantum theories (Olson and Fiol
2001).
Previous work on this subject analysed the projectile deflection for He ionization by
100 MeV u−1 C6+ and 3.6 MeV u−1 Au53+ . Good agreement between the theoretical and
experimental results was found for C6+ , but serious discrepancies arose for the high perturbation
Au53+ case. The calculations show that, besides the inclusion of the internuclear interaction,
the choice of the effective three-body model for the description of the four-body target is
crucial in the quantum description of the experimental results. Furthermore, complications
due to the double ionization of He have been addressed via classical trajectory Monte Carlo
(CTMC) calculations. In order to avoid these two major difficulties, in this work we study the
mechanisms of ionization for an atomic hydrogen target. This choice leads to a real three-body
system with pure Coulomb interactions between all particles. A systematic study on atomic
hydrogen ionization as a function of projectile momentum transfer allows one to elucidate threebody collision mechanisms and predict electron spectra signatures due to various dynamical
processes.
Three-body dynamics in hydrogen ionization by fast highly charged particles
1761
Single ionization of ground and n = 2, 3 excited state hydrogen by 3.6 MeV u−1 C6+ and
Au are investigated. CTMC and quantum mechanical continuum distorted wave (CDW)
theories are employed. Atomic units are used throughout this work (me , e, h̄ = 1). In section 2,
the theoretical methods are briefly described. Next, the ionization of ground and excited states
of hydrogen by C6+ are studied. In section 4, the ionization process in the non-perturbative
regime (Au53+ + H) is investigated. In section 5, a detailed study of the collision dynamics and
its interpretation in terms of classical collision mechanisms is given.
53+
2. Theoretical approaches
We have investigated hydrogen ionization by the impact of a charged particle. The projectiles,
C6+ and Au53+ are considered point charges, neglecting the tightly bound electrons in the Au53+
case. The calculations were performed within two very different approximations, namely the
CTMC and quantum mechanical CDW theories. In both theories the internuclear interaction is
included along with the projectile–electron and target–electron interactions. As will be shown,
the inclusion of all interactions is essential for the proper account of the momentum transferred
to the target system during the collision.
The hydrogen atom was chosen for the present study because it can be properly treated
in both the quantum mechanical and classical theories since only Coulomb interactions are
present. Moreover, it is well known that a classical microcanonical ensemble yields the exact
quantum mechanical momentum density (Fock 1935, Abrines and Percival 1966)
8pn5
ϕ̃nm (p)2 =
ρ̃n (p) =
.
(1)
π 2 (p 2 + pn2 )4
,m
However, the classical spatial density is too compact compared to the quantum one and vanishes
identically beyond r = 2 au. In the three-body CTMC method employed here the initial state
of the H(1s) atom is described by means of a modified Wigner distribution (Hardie and Olson
1983, Wood et al 1997). This method utilizes a discrete superposition of microcanonical
ensembles in order to provide a good approximation to the H(1s) initial radial distribution.
This description of the initial state slightly degrades the momentum distribution. The choice
of such a method over the single microcanonical ensemble approach is grounded on the fact that,
as will be shown below, the results are strongly dependent on the spatial electron distribution.
Furthermore, this method has been extensively tested for a wide range of energies and projectile
incident charge states and provides excellent agreement with experiment (Illescas and Riera
1999).
The CTMC method is well documented (Olson and Salop 1977, Reinhold and Olson
1989). The initial conditions are set such that the projectile impinges with a velocity v and
a randomly chosen impact parameter b on the atomic target from a large distance, such that
the interaction between the two fragments is negligible. The classical equations of motion are
solved for the Hamiltonian
ZP
ZP
1
−
−
(2)
H = H0 +
rN
rP
rT
where H0 is the kinetic energy, and the coordinates rN , rT are the positions of the projectile
and electron relative to the target nucleus, and rP is the position of the electron relative to the
projectile P of charge ZP .
The CTMC method has successfully described atomic ionization collisions by heavy and
light projectiles on a wide range of incidence energy for a large number of atomic and molecular
targets (Schultz and Olson 1988, Olson 1996). In particular, it has recently been applied to
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J Fiol and R E Olson
describe electron momentum spectra for helium ionization by impact of 3.6 MeV u−1 Au53+
(Schmidt et al 1998) and to the helium triple differential ionization cross section (TDCS) for
100 MeV u−1 C6+ collisions (Schulz et al 2001a).
In this work we are interested in the evaluation of the TDCS and DDCS
dσ
dσ
dσ
and
=
de .
(3)
dQ⊥ dEe de
dQ⊥ dEe
dQ⊥ dEe de
Here Ee and e are the ionized electron energy and solid angle, and Q⊥ is the perpendicular
component of the momentum transferred to the atom by the projectile Q = MP v − K , where
MP , v and K are the projectile’s mass, initial velocity and final momentum, respectively.
Since the final projectile momentum is very close to the initial (Q K) the scattering angle
is related to the perpendicular momentum transfer by Q⊥ ≈ MP vθ . Thus, the corresponding
cross sections are trivially related by
dσ
dσ
(MP v)2
≈
.
dEe de dP
2πQ⊥ dQ⊥ dEe de
The quantum mechanical calculation of the TDCS is performed through
(2π)5
dσ
=
kQ⊥ |tif |2 ,
dQ⊥ dEe de
v2
tif = #f− |Vf |#i (4)
where Q = (Ee + |εi |)/v and εi is the atomic binding energy. Here tif is the transition matrix
element between the initial and final states. The final state is chosen to be the C3 or CDW
wavefunction (Garibotti and Miraglia 1980) that is given by
#f± (r, R) =
ei (kT ·rT +KT ·RT ) ±
D (νT , kT , rT )D ± (νN , kN , rN )D ± (νP , kP , rP )
(2π)3
(5)
where
D ± (να , kα , rα ) = )(1 ± iνα ) e−π να /2 1 F1 [∓iνα ; 1; −i(kα rα ∓ kα · rα )].
In (5) να is the Sommerfeld parameter for the Coulomb interaction Zα /rα between a given pair
of particles, kα is their relative momentum, and mα is the corresponding two-body reduced
mass. Rα , Kα are the position and the momentum of the remaining particle relative to
the pair. The simplest description of the initial state is made through the unperturbed Born
wavefunction (B1)
#iB (rT , RT ) =
ei KT ·RT
φi (rT )
(2π)3/2
(6)
that includes the motion of the projectile and the bound state of the target. In this work we
present calculations with the Born initial state (CDW-B1) and with the eikonal initial state
(CDW-EIS, see Crothers and McCann (1983), Fainstein et al (1991))
# EI S (rT , RT ) = #iB (rT , RT ) eiνP ln(kP rP −kP ·rP ) eiνN ln(kN rN −kN ·rN ) .
(7)
The difference between our CDW-EIS calculations and the standard CDW-EIS approximation
is that, because we are interested in the scattering of the projectile, the internuclear interaction
has not been neglected. We note that the internuclear interaction has previously been included
for small scattering angle in both FBA and CDW-EIS theories in a semiclassical approximation
by means of a phase factor (Salin 1989, Rodrı́guez 1996). In our calculations the three
interactions have been taken into account on exactly the same footing in the wavefunctions
and perturbation potential (Fiol et al 2001).
Three-body dynamics in hydrogen ionization by fast highly charged particles
1763
-15
10
6+
3.6 MeV/u C + H(1s)
-16
Ee = 10 eV
2
dσ/dQ⊥dEe (cm /a. u.)
10
-17
10
50 eV
-18
10
90 eV
-19
10
0.1
1
2
3 4 5
Q⊥ (a. u.)
Figure 1. DDCS as a function of the projectile transverse momentum transfer at fixed electron
energies for ionization of hydrogen by 3.6 MeV u−1 C6+ . Curves: CDW-B1 calculations; and
symbols: CTMC results, which are convoluted over the transverse momentum resolution of
/Q⊥ = ±0.1 au and the electron energy /Ee = ±10 eV.
The final distortion potential Vf in the transition matrix element (4) for the CDW
wavefunction (5) can be written as
∇D −∗ (νj , kj , rj )
KP KN
KT KN
(8)
Vf =
−
− KT KP
Kj =
MP
MT
D −∗ (νj , kj , rj )
where MP and MT are the masses of the projectile and target nucleus, respectively (Fiol and
Olson (2002) and references therein). The above expression is valid for any relation of masses
between the fragments. In the limit of infinitely heavy nucleus (MP , MT 1) only the last
term contributes and the usual CDW result is recovered.
3. Perturbative collisions
Very recently, single ionization of helium produced in collisions with 100 MeV u−1 C6+ and
3.6 MeV u−1 Au53+ was studied by investigating the DDCS dσ/dQ⊥ dEe as a function of
the ionized electron energy and perpendicular momentum transfer (Moshammer et al 2001,
Olson and Fiol 2001). Although the experimental data presented similar behaviour for both
projectiles, the theoretical analysis showed marked differences. The calculations revealed that
for the Au53+ + He system four-body effects are important. Moreover, the shape of the cross
section was shown to strongly depend on the inclusion of the internuclear interaction and the
choice of the model potential for the He+ core. Additionally, the CTMC calculations indicated
that double ionization dominates over single ionization for large values of the momentum
transfer. These problems are absent in calculations for the atomic hydrogen target.
In this section we analyse the hydrogen ionization cross section (3) for 3.6 MeV u−1
6+
C impact (figure 1). The C6+ system is in the perturbative regime, νP = 0.5, which
is characterized by close projectile–electron collisions. The projectile primarily transfers
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J Fiol and R E Olson
6+
3.6 MeV/u C + H(n)
-17
n=1
2
2
dσ/dQ⊥dEe (cm /a. u. )
10
-18
10
n=2
n=3
-19
10
0.
2
3 45
Q⊥ (a. u.)
Figure 2. Comparison of the DDCS for the ground, n = 2 and 3 excited states of hydrogen.
The electron emission energy is 50 eV; the resolutions in the CTMC calculations are the same as
in figure 1.
momentum to the electron without a strong interaction with the nucleus target, leading to a
readily apparent binary peak. The strong binary peak is the result of the collision distance
between the projectile and the electron being smaller than the characteristic radius of the atom
in the initial state.
In two-body collisions with Coulomb interactions the perpendicular component of the
momentum transferred by the projectile is given by
2νj
j
Q⊥ =
(9)
bj
where bj is the two-body impact parameter and νj is the corresponding Sommerfeld parameter.
For instance,
√ with an electron energy Ee = 50 eV the corresponding momentum transfer is
Qe = k = 2(Ee + |εi |) ≈ 2.16 au. The component parallel to the initial velocity is fixed by
energy conservation
Ee + |εi |
Q =
,
(10)
v
which for this case is Q = 0.15 au. Thus, the projectile must approach to a distance of
be ≈ 0.46 au with the electron. This characteristic close projectile–electron collision is
even more pronounced for higher electron energy emission. For ionized electron energies
Ee = 90 eV, the relative impact parameter decreases to be ≈ 0.36 au.
The projectile–electron impact parameter be must be compared with the radial expectation
value of the atom r0 . For the ground state this radius is r0 1s = 1.5 au, while the expectation
values of the hydrogen atom in the n = 2 and 3 excited states are r0 n=2 = 5, 6 au for = 0,
1 and r0 n=3 = 10.5, 12.5, 13.5 au for = 0, 1, 2, 3, respectively. The distance of closest
approach between the projectile and the target nucleus bR ≈ b ≈ r0 ± be increases with
the quantum number n. Thus, according to (9) the momentum transferred to the recoil ion
decreases and binary-encounter collisions dominate. Shown in figure 2 are the cross sections
Three-body dynamics in hydrogen ionization by fast highly charged particles
1765
for ionization of the n = 2, 3 excited states. The initial n = 2 and 3 states were described by
means of a microcanonical ensemble of energy En = −1/2n2 in the CTMC calculations. In
order to perform a fair comparison between the two theories, the CDW results for n = 2 have
been averaged over the 2s and 2p 0, ±1 states. Quantum mechanical calculations for the n = 3
initial state are not presented because their computing times are prohibitive within the present
approximations.
The cross sections for ionization of the excited states of hydrogen in figure 2 are strongly
peaked around Q⊥ ≈ k accordingly with the simple projectile–electron binary collision picture.
In fact, the binary peak width for the excited states is mainly determined by the electron’s
Compton profile, since the target nucleus is far removed from the interaction region. For such
perturbative collisions a laser excited target will present the opportunity to study collisions of
ions with essentially a free-electron gas target.
The study of the ionization of excited states of hydrogen does not only present theoretical
interest. Although the experimental production of the atomic hydrogen target in either ground
or excited states is difficult, the cross section for ionization of alkali atoms is closely related to
the present calculations. These atoms, having only one active electron, are expected to present
the same qualitative behaviour as atomic hydrogen. It has recently been possible to produce a
cold target of alkali atoms by means of magneto-optical traps (MOT). This technique allows
one to study collisional processes involving both ground and excited initial states (van der Poel
et al 2001, Turkstra et al 2001, Flechard et al 2001).
In order to study in more detail the mechanisms of ionization we have plotted in figure 3
the TDCS dσ/dQ⊥ dEe de in the plane defined by the initial velocity of the projectile
and the perpendicular momentum transfer vector Q⊥ . We have chosen four values of
Q⊥ = 0.5, 1, 2, 3 au. These values correspond to regions below, at and above the binary
collision condition Q ≈ k for 50 eV electrons. For all cases the electrons are emitted along the
direction of the momentum transfer (marked with the arrows in the plots). For Q⊥ = 0.5 and
1 au the recoil ion momentum is directed opposite to the direction of the electron emission. In
these cases the electron and the recoil ion are emitted back-to-back, the recoil ion balancing a
fraction of the momentum of the ionized electron since the total momentum transfer is fixed.
For values of Q⊥ equal to, or greater than, the electron momentum, the recoil ion is
constrained by momentum conservation KR = Q − k to the angular range θR− < θR < θR+ ,
where θR± = θQ ± sin−1 (k/Q) and θQ is the polar angle of the momentum transfer Q. For
instance, for Q⊥ = 2 au the recoil must lie in the half of the plane given by the Q⊥ vector.
For Q⊥ = 3 au such a restriction is more evident as the recoil momentum angle is limited
to a narrow range, 46◦ < θR < 126◦ . This kinematic constraint gives rise to singularities in
the recoil ion spectra. In figure 3 the CDW calculations clearly display ‘wings’ for Q⊥ = 2
and 3 au at the positions where several electron angles contribute to one recoil ion angle. The
CTMC calculations also display the same tendency, but because of the finite resolution used
for the electron energy, momentum transfer and collision plane azimuthal angle, the ‘wings’
are smoothed. The CDW predictions are clearly confirmed by the CTMC calculations.
4. Ionization by highly charged ion impact
In figure 4 are shown the DDCSs for ionization of ground state hydrogen by 3.6 MeV u−1
Au53+ . The shapes of the DDCS differ from those observed for C6+ at the same electron energies
and values of momentum transfer. The Au53+ system shows a dominance of low momentum
transfer collisions. In contrast, binary-encounter electrons portrayed by the so-called Bethe
ridge characterized the spectra in the C6+ case. However, neither the CTMC nor the CDW-B1
calculations display a strong contribution in the binary peak region for Au53+ impact. These
1766
J Fiol and R E Olson
90
3
120
90
60
120
2
30
150
0 180
0
1
0 180
0
3
210
Q⊥ = 0.5 a. u.
2
330
210
Q⊥ = 1.0 a. u.
330
6
2
2
TDCS (10 cm /a. u. sr)
30
150
3
1
240
3
300
240
-18
270
120
300
270
90
30
90
1.5
60
120
60
1.0
20
30
150
30
150
0.5
10
0 180
0
10
0.0 180
0
0.5
210
Q⊥ = 2.0 a. u.
20
30
60
6
240
300
270
330
210
Q⊥ = 3.0 a. u.
1.0
1.5
240
330
300
270
Figure 3. TDCS for ionization of H by 3.6 MeV u−1 C6+ in the plane defined by the initial velocity
and the perpendicular momentum transfer Q⊥ . CTMC (symbols) results for fixed values of the
projectile transverse momentum and electron energy are compared to CDW-B1 (lines) as a function
of the electron (solid symbols and lines) and recoil (open symbols and dashed lines) angles. The
CTMC resolutions were set to /Ee = ±10 eV, azimuthal angle /ϕ = ±20◦ and /Q⊥ = ±0.1 au.
results for atomic hydrogen mirror those previously presented for 3.6 MeV u−1 Au53+ + He
single ionization (Olson and Fiol 2001).
The interpretation of the Au53+ + H(1s) behaviour, νP = 4.42, as compared to the
perturbative C6+ case, νP = 0.5, follows the explanation given for the helium target in
our previous paper. For coulomb collisions the Au53+ projectile–electron collision distance
(equation (8)) for electrons emitted with 50 eV (Q⊥ ≈ 2.15 au) is be ≈ 4.1 au, which unlike
in the C6+ case is considerably larger than the radius of the atom. The nuclear–nuclear impact
parameter in such collisions is estimated to be about 4.1 ± 1.5 au. For these distances, the
momentum transferred from the projectile to the target nucleus is also large and ranges from
1.5 to 3.5 au with the recoil ion momentum vector pointed opposite to that of the ionized
electron. Vector addition of the electron and recoil ion momenta results in small values of
overall momentum transfer. Thus, we find that Au53+ collisions strongly involve all three
bodies and cannot simply be considered as a binary encounter collision between the projectile
and electron as in the C6+ case.
Three-body dynamics in hydrogen ionization by fast highly charged particles
1767
-13
10
3.6 MeV/u Au
+ H(1s)
Ee = 10 eV
-14
10
50 eV
2
dσ/dQ⊥dEe (cm /a. u. )
53+
-15
2
10
90 eV
-16
10
-17
10
-18
10
0.
2
3 4 5
Q⊥ (a. u.)
Figure 4. The same as in figure 1 except that the projectile is now 3.6 MeV u−1 Au53+ .
It is possible that collisions with highly charged projectiles can induce polarization of the
initial state leading to projectile electron collisions at larger impact parameters than those given
here (Moshammer et al 2001). However, one can argue that the collision time is too short
for polarization of the target to be important. The collision time is approximately given by
τ = 2b/v. For an impact parameter of 1.5 + 4.1 au, this leads to a collision time of 0.93 au. The
orbital period of H(1s) is approximately 9.4 au. Thus, as in the He case we expect insignificant
polarization of the target before the collision. Although the CDW quantum mechanical
approach employed in this work may lack these polarization effects, they are included in the
CTMC treatment (see, for example, Olson (1979)). The excellent agreement between theories
observed in figure 4 indicates that polarization effects are not important in this system.
From our discussion it is apparent that the underlying physics of these collisions is
determined by the magnitude of the radial expectation value of the atom compared to the
pairwise ranges of interaction. In figure 5 we show the average impact parameter b
from our
CTMC calculations for the two collision systems as a function of the perpendicular component
of the momentum transfer at the three electron energies investigated, Ee = 10, 50 and 90 eV.
The impact parameter of C6+ is observed to be comparable to r0 . Moreover, at 50 and 90 eV
we observe a peak in the impact parameter distribution for momentum transfer corresponding
to the binary peak position. Using equation (9), it is clear that the recoil ion has a minor
contribution in the binary peak region for the C6+ system. On the other hand, the Au53+ impact
parameters do not possess maxima and are considerably larger, confirming that both the ionized
electron and recoil ion participate in the overall momentum balance.
In order to assess the importance of the relative distances between particles as compared
to the range of interactions, we display in figure 6 the ionization cross sections for the n = 1, 2
and 3 states of hydrogen. The shape of the DDCS for Au53+ impact on the excited states tends
to that predicted for the C6+ perturbative case as the radius of the atom becomes larger than the
range of the interaction. For instance, for 50 eV the projectile–electron distance be ≈ 4.1 au
1768
J Fiol and R E Olson
10
53+
< b > (a. u.)
Au
1
6+
C
10 eV
50 eV
90 eV
0.1
0.
3 4 5
2
Q⊥ (a. u.)
Figure 5. Average impact parameter for H(1s) ionization by 3.6 MeV u−1 Au53+ and C6+ .
-14
10
3.6 MeV/u Au
53+
+ H(n)
-15
10
n=2
2
2
dσ/dEe dQ⊥ (cm /a. u. )
n=1
-16
10
n=3
-17
10
0.
2
3 4 5
Q⊥ (a. u.)
Figure 6. Comparison of DDCS for the Au53+ + H(n) system with n = 1, 2 and 3 for 50 eV
electron emission. Solid curves represent the CDW-B1 calculations; and the symbols represent the
CTMC results. The CTMC resolutions are as in figure 1.
is much smaller than the size of the atom in the n = 3 excited state r0 n=3 ≈ 12 au. Thus,
the momentum transferred to the nucleus ranges from 0.5 to 1 au, similar to the C6+ + H(1s)
case. For Au53+ impact the excited states show a small shoulder at Q⊥ ≈ 3 au, the origin of
which will be discussed in the next section.
Three-body dynamics in hydrogen ionization by fast highly charged particles
1769
5. Collision dynamics
Our study of hydrogen ionization by C6+ and Au53+ impact indicates that the range of the interaction as compared to the atomic spatial distribution determines the overall shape of the DDCS.
The marked differences in the results for ground and excited states of hydrogen support this idea.
In this section the different collision dynamics that lead to the observed spectra are discussed.
A detailed investigation of the three-particle dynamics can be performed through the study
of kinematically complete cross sections (Schulz et al 2001a, 2001b). Figure 7 shows the Au53+
TDCS for electrons emitted at 50 eV in the plane defined by the v and Q vectors. For small
values of Q⊥ the electrons are emitted along the direction of the momentum transfer, i.e. at an
angle θQ associated with the Q vector. As expected, in these collisions the recoil ion is found
along the −Q vector. Such angular distributions are in accord with a binary-encounter collision
picture. As the perpendicular momentum transfer is increased, a second peak emerges at an
angle 360◦ − θQ . With a further increase in momentum transfer this second peak dominates
the spectra, as is shown in the right-most plots in figure 7 for Q⊥ = 1.6 au. This tendency
is confirmed for higher perpendicular momentum transfer values. This behaviour is quite
surprising since it has never been observed for other systems. In particular, the TDCS for
C6+ impact, plotted in figure 3, does not present a similar trend. Moreover, the transition for
electron scattering at angle θQ to the angle 360◦ − θQ occurs before one reaches momentum
transfers associated with binary peak scattering, Q⊥ = 1.9 au.
Our CDW and CTMC calculations indicate that the transition between the two different
angular distributions occurs quite rapidly. Both theories show that the distribution switches
between both regimes in a narrow interval /Q⊥ ≈ 0.2 au. However, there is a discrepancy in
the value at which the transition takes place. For the CTMC results the value of the transition is
Q⊥ ≈ 1.1 au, while in the CDW-B1 and CDW-EIS models it is slightly higher, Q⊥ ≈ 1.2 au.
The angular distributions presented in figure 7 are related to different collision
mechanisms. The peak observed near to the angle θQ of the momentum transfer Q is produced
in a distant collision as shown in figure 8 (a). This kind of collision is similar to the binaryencounter mechanism in that the projectile–electron interaction is stronger than that between
the two nuclei. This mechanism is the main one responsible for ionization in the C6+ case. Its
signature is the electron spectra peaking near θQ with the recoil ions found scattered back-toback at θQ + 180◦ .
A second collision mechanism is displayed in figure 8(b) with the projectile penetrating
the electron cloud. These are related to small impact parameter collisions resulting in large
momentum transfer, and contribute in the C6+ case for Q⊥ > k, but are not observed in the
Au53+ + H(1s) system because the range of interaction exceeds the radial dimensions of the
target atom. However, by increasing the atomic dimensions this kind of collision produces
the peak observed in the DDCS (figure 6) near Q⊥ = 3 au in the Au53+ ionization of the
n = 3 state of hydrogen. Here, the collision distance and the momentum transfer between
the projectile and the electron are approximately be = 4 au and Qe⊥ ≈ 2.1 au. For collision
8(b) the distance between nuclei can be estimated to bR = r0 n=3 − be ≈ 8 au. Thus, the
momentum transfer to the target ion is QR⊥ ≈ 1 au. In this case both the electron and the recoil
scatter in the same direction so that the net momentum transfer is now Q⊥ ≈ 3 au.
Figure 8(c) shows a third mechanism where the projectile suffers a stronger interaction
with the recoil ion than with the electron. After the ionization takes place the electron is forced
to swing by its parent nucleus. We observe that this mechanism only contributes when the
range of the interaction is larger than the size of the atom.
It is important to note that in the case of ‘swing by’ electrons, neither the electron nor the
recoil ion is emitted along the direction of momentum transfer (see Q⊥ = 1.6 au in figure 7).
1770
J Fiol and R E Olson
90
-18
2
TDCS (10 cm /a. u.)
450
120
90
450
60
120
300
300
30
150
150
30
150
150
Q⊥= 0.8 ± 0.1 a. u.
0 180
Q⊥ = 0.8 a. u.
0
150
0 180
0
150
330
210
300
330
210
300
240
450
300
240
450
270
120
90
45
60
-18
120
60
30
30
150
15
0 180
30
150
15
Q⊥ = 1.1 ± 0.2 a. u.
2
TDCS (10 cm /a. u.)
30
Q⊥ = 1.2 a. u.
0
15
0 180
0
15
330
210
30
330
210
30
240
45
300
240
45
270
120
300
270
90
24
90
24
60
16
120
60
16
30
2
150
8
-18
TDCS (10 cm /a. u.)
300
270
90
45
0 180
30
150
8
Q⊥ = 1.6 ± 0.3 a. u.
Q⊥ = 1.6 a. u.
0
8
0 180
0
8
330
210
16
24
60
330
210
16
240
300
270
24
240
300
270
Figure 7. Classical CTMC and quantum mechanical CDW TDCSs for ionization of ground state
hydrogen by Au53+ impact to produce 50 eV electrons. In the left column the CTMC results
for the electron (solid squares) and the recoil ion (open circles) were convolved over the energy
/Ee = ±10 eV, and azimuthal angle /ϕ = ±20◦ resolutions. The resolutions in the perpendicular
momentum transfer are specified in the plots. In the right panel the CDW-B1 (electron: full lines,
recoil: dashed lines) and CDW-EIS (symbols as in CTMC) results are displayed.
Three-body dynamics in hydrogen ionization by fast highly charged particles
MP v
+
(a)
(b)
KR
Q
k
1771
+
+
-
- k
MP v
Q
+
KR
-
k
KR
(c)
+
MP v
+
Q
Figure 8. Schematic diagram of the three ionization mechanisms leading to (a) binary electrons,
(b) three-body collisions and (c) swing by electrons.
6
5
x (a. u.)
4
t2
t1
t0
t2
3
"binary electrons"
2
t1
1
0
t0
-1
5
t2
4
"swing-by electrons"
x (a. u.)
3
2
1
t1
t0
t2
0
-1
-2
t1
t0
-5 -4 -3 -2 -1 0 1
z (a. u.)
2
28 29 30
Figure 9. CTMC trajectories illustrating the mechanisms that produce binary electrons (upper
plot) and swing by electrons (lower plot). The projectile follows an almost straight line trajectory
and the recoil ion remains approximately at rest. The horizontal axis, displaying the component
of the position of the particles parallel to the initial velocity is approximately proportional to the
time. The projectile scattering plane defines the x component.
The signature of this mechanism is observation of electrons emitted at 360◦ −θQ with the recoil
ion being found peaked near 180◦ − θQ . Note, these are not the recoil electrons commonly
found in electron impact ionization that peak at 180◦ + θQ .
1772
J Fiol and R E Olson
To confirm our interpretation of the collision mechanisms, in figure 9 we show CTMC
trajectories illustrating the ‘binary’ and ‘swing by’ electron dynamics for Au53+ + H(1s). In
the upper plot, the binary electrons are produced in distant collisions (b ≈ 5 au). On the
other hand, the swing-by electrons are produced in closer collisions, with impact parameter
b ≈ 2 au. The projectile passes on one side of the target nucleus while the electron is on the
opposite side. The direction of the perpendicular momentum transfer (not distinguished in the
figure) is determined by the interaction with the recoil ion, which is opposite to the case for
binary electrons.
6. Conclusions
Ionization cross sections for fast ion–atom collisions have been calculated in the perturbative
(C6+ , νP = 0.5) and non-perturbative (Au53+ , νP = 4.4) regimes. In general there is excellent
agreement between calculations made using the CTMC and CDW-B1 and CDW-EIS theories.
This agreement extends to doubly differential and kinematically complete TDCSs.
Three collision mechanisms have been identified. The range of the projectile–electron
interaction (collision distance) as compared to the size of the atom determines the relative
importance of each of them. The first process is the well known binary encounter between the
projectile and electron whose signature is the Bethe ridge. This mechanism is compromised
for the case of highly charged ions impact because the long-range internuclear interaction
allows the recoil ion to also actively participate in the collision. Because of the long range of
interaction compared to the orbital dimensions of the target atom, no Bethe ridge is found for
3.6 MeV u−1 Au53+ + H(1s) collisions.
In contrast, binary-like electrons are found for high incident velocities, even in nonperturbative collisions, when the size of the atom is larger than the collision distance. This
was verified by studying the ionization of n = 2 and 3 excited states of hydrogen. Here, a
second mechanism also occurs when the projectile penetrates the electron cloud interacting
simultaneously with the recoil and the electron. Such collisions result in large momentum
transfer and can give rise to structures on the DDCS. A laser-excited alkali atom target could
provide the framework for the experimental study of these kind of processes.
The third mechanism results in our prediction of ‘swing by’ electrons. For strong nonperturbative collisions where the range of interaction is greater than the target atom’s orbital
dimensions, it is possible for the projectile to closely interact with the recoil ion while the
electron is forced to swing by its parent ion. This mechanism leads to the ionized electron and
recoil ion spectra not being found peaked parallel to the momentum transfer vector, but in a
mirror position to it. To date these electrons have not been observed.
Acknowledgment
The authors acknowledge the support from the Office of Fusion Energy Sciences, DOE.
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