Excitation-ionization of helium in high energy collisions
A. L. Godunov*, C. T. Whelan*, and J. M. McGuire†
*Department of Physics, Old Dominion University, Norfolk, VA 23529-0116
†
Department of Physics, Tulane University, New Orleans, LA 70118-5698
Abstract. We review recent developments in the process where excitation occurs together with ionization in collisions
of helium with charged particles and photons. Various aspects of this process are discussed. The interrelation between
cross section ratios for charged particle and photon impact is considered in the limit of first order perturbation.
collision Coulomb interactions (PCI), final state
correlation (FSC), distorted wave methods (DWBA),
the two step one (TS1) mechanism (and related
Corkum mechanism for strong photon fields) as well
as the general issue of mechanisms for correlation
dynamics all involve correlation between continuum
virtual states in excitation-ionization. In addition the
validity of the widely used closure approximation has
recently been brought into question in some specific
cases of excitation-ionization.
This leads to
observable
effects
of
energy-non-conserving
intermediate states, time ordering of the external
interactions, and time correlation between different
electrons [6]. One may also compare and interrelate
cross sections for impact by electrons, protons,
positrons, antiprotons, ions, photoannihilation and
Compton scattering.
1. INTRODUCTION
The process of excitation-ionization provides a
relatively efficient way to study the physics of few
electron systems since two electrons undergo a
transition. A single active electron approach without
correlation is completely unsatisfactory since it yields
zero transition probabilities. At very high energies,
where first order perturbation theory is applicable,
excitation-ionization is dominated by dynamics of
electron-electron correlation since without coupling
between electrons only one electron can make a
transition since there is only a single interaction with
the perturbing projectile. At moderately high energies
the interaction between the atom and the incident
projectile may be treated in some form of second order
perturbation theory. Because this process does not
occur in first order in the absence of electron
correlation, the second order contribution is significant
over a relatively wide range of collision energies and
depends strongly on the dynamics of electron
correlation. In this paper we consider both the
moderately high energy second Born regime and then
the very high energy first Born regime.
2. FORMULATION
In excitation-ionization a five fold differential cross
section is observable. This may be expressed in terms
of the scattering amplitude, f, namely,
(2π ) k s k f
d 5σ
f
=
dΩ s dΩ f dEs
4k 0
Here we consider several aspects of excitationionization. Perhaps the most important of these are
new studies of electron correlation [1-5].
Understanding correlation in continuum states has
been a challenge to atomic physics for many years.
This continuum correlation is clearly a significant part
of the total contribution of electron-electron
correlation that constitutes a major portion of
excitation-ionization cross sections. The role of post
4
2
(1)
where s denotes the outgoing electron and f denotes
the scattered particle. Following Whelan [1], the
scattering amplitude may be written,
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© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
32
f = Ψ − | V | Φ i = Φ f | V ∑ (G0−V ) | Φ i
n
where <f| and <i| are the initial and final states of the
helium atom and q = |kf - k0| is the momentum transfer
of the projectile. Note that fB1 = 0 for an uncorrelated
system where <f| = <1f|<2f| and <i| = <1i|<2i|. In a
correlated atom the first Born amplitude is finite, but
small. Hence for two electron transitions there is often
a wide range of energies where the second Born
approximation is valid. At very high energies the first
Born term eventually dominates.
(2)
n
Here Ψ− is the wavefunction to the full Hamiltonian
satisfying incoming boundary conditions, and Φi and
Φf are the unperturbed initial and final wavefunctions.
The Greens function propagator,
G0− =
1
E − Ei − iη
gives the propagation in energy space. Its Fourier
transform gives propagation in time.
Incoming
boundary conditions are used so that attention can be
focused on Ψ−, where electron correlation in the final
state can more easily be examined in detail.
2.1 Relation between charged particle and
photon cross sections
In this very high energy regime it is possible to
relate cross sections for charged particles and photons.
This occurs because the scattering amplitude for
photons and charged particles both contain matrix
elements similar to the atomic form factor,
<f|exp(iqr)|i>, in Eq. (6) above. If the photon is
inelastically scattered, but not annihilated, the process
is called Compton scattering [8] and the corresponding
scattering amplitude has a different overall constant
and no factor of (Z/q)2, arising from Coulomb
scattering between the charged particle in Eq. (6).
Then the matrix elements are identical and the cross
section for Compton scattering and charged particle
scattering are simply related [9]. If the photon is
annihilated, the process is photoannhilation [3],
corresponding to the Einstein photoeffect.
The
amplitude for this process is proportional to
<f|ε⋅rexp(iqr)|i>, where ε is the unit vector of the
electric field of the photon. When qr < 1 the cross
section for photoannhilation is usually much larger,
especially in the dipole limit where qr << 1. In this
dipole limit the amplitudes for charged particle
scattering, Compton scattering and photoannhilation
are all proportional to the dipole matrix element
<f|z|i>.
Of course the cross section for
photoannhilation is kinematically more restricted than
for either Compton scattering or charged particle
scattering, where in the final state the incident particle
can scatter into a variety of different directions. This
well-known dipole limit, known as the Bethe limit, is
valid for dipole transitions for charged particle
scattering only at extremely high energies.
In the high energy regime the Born series in V may
be truncated at the second term so that,
B1
f ~ f
+f
B2
=
Φ f V +V
1
Φi
E − Ei − iη
(3)
The most challenging part of these calculations is
evaluation of the second order amplitude with fully
correlated intermediate states, namely [7],
lim ∑ d 3k
η →0+
α
r
(4)
r
r
r
φ f | V ( k − k f ) | φα φ f | V (k i − k ) | φα
r r
r r
( k − k f ) 2 [k i2 − k 2 − 2m p Eα + iη ]( k i − k ) 2
This energy propagator is separated into two terms,
1
=
k − k − 2m p Eα + iη
2
i
(5)
2
iπδ ( k i2 − k 2 − 2m p Eα ) −
PV
k − k − 2m p Eα
2
i
2
The principal value (Pv) contribution is the most
difficult to evaluate. The numerical integration is time
consuming. Omitting this term typically reduces the
computational time by over two orders of magnitude
in our calculations.
These interrelations between scattering by charged
particles and photons hold for multiple electron
transitions as well as for single electron transitions.
This leads to a useful simplification, namely a relation
between the ratio, R of double to single electron
ionization cross sections [3]. For excitation-ionization
this leads to,
At very high energies for projectiles of charge Z
the first Born approximation to the scattering
amplitude is given by,
f
B1
=−
4Z 2
r r
< f | exp(iq ⋅ r ) | i >
2
q
(6)
33
R=
σ Z+* ∞ d
= ρ (∆E ) Rγ A (∆E ) +
σ Z* ∫I Z
∞
∫ρ
nd
Z
the construction of analytic forms for the continuum
three-body wavefunction [13]. It is now well known
that such methods give more accurate shapes than
magnitudes in most cases for differential cross
sections. A more accurate approach seems to be the
use of numerical methods, such as close coupling
methods [1,4,5] to evaluate Ψ−. The simplest term
contribution to Ψ− in many body perturbation theory is
the two step one (TS1) term, which corresponds to
Coulomb scattering of one electron off a second
electron following a single interaction with the
projectile. This rather simple model, which gives a
perhaps overly simple qualitative picture for electron
correlation dynamics, has nonetheless been widely
used to describe a large amount of data, especially
under conditions of weakly perturbed systems. An
extension of this TS1 model has been used to describe
how a two-electron transition may occur in a strong
photon field. In this model, often called [14,15] the
Corkum model, the strong photon field causes the first
electron to return to the target atom where it interacts
with the second electron. This may be thought of as
TS1 in an atomic state dressed by the photon field.
(7)
(∆E ) Rγ C (∆E )
I
where
ρ Zj =
1
σ Zj*
dσ Zj / d∆E is the density of states
for dipole and non dipole contributions to single
ionization by a particle of charge Z and
Rγk = σ γ+k* / σ γ+k
for
photoannhilation
(A)
and
Compton scattering (C). Here I is the energy threshold
for ionization-excitation and ∆E is the energy
transferred to the target electron. We expect, as in
double to single ionization, that the first term on the
right of Eq.(7) will tend to dominate and that all
multipoles may be simply used in ρZ. This gives a
simpler expression that is easier to apply to data.
Eq.(7) has been used for ratios of single to double
ionization to check the inter consistency of photon and
charged particle data [10]. However, for excitationionization none of these ratios has yet been determined
either theoretically or experimentally, although
experiment and theory can both now be done. The
photon experiments require photon energies up to tens
of keV. Such photons are becoming available with
new synchrotron radiation sources [11].
The closure approximation is widely used [1,5] in
the evaluation of cross sections for excitationionization. Godunov et al. [16] have tested this
approximation by numerically evaluation the principal
value term in Eq. (5) and have shown that in many
cases this closure approximation is justified. However
there are cases where closure is not accurate. The
principal value term of Eq. (5) can be transformed
from energy to time space. This term corresponds to
time ordering of the interactions in time. If the
interactions correspond to interactions of the projectile
with different electrons, the contribution from this
principal value term produces correlation between the
time evolution of different electrons. In the absence of
such effects, one has independent time evolution of
electrons. This limit, corresponding to the closure
approximation, has been referred to [17] as the
independent time approximation (ITA). Deviations of
about 30% from the independent time approximation
have been observed in the polarization of light emitted
from the n=2 level of helium following ionization with
excitation into n=2 by impact of high energy protons
and electrons [18].
3. DISCUSSION
New effects of electron correlation have been a
primary focus of attention in recent studies of two
electron transition cross sections in helium [1,3].
Perhaps the simplest and best understood correlation
effect is that of correlation in bound states, which has
been studied for many years [12]. Such effects are
relatively straightforward to include in calculations for
excitation-ionization. This includes the effect of
ground state correlation (GSC), which corresponds to
one of the lowest order terms in many body
perturbation theory that have been used to characterize
two electron transitions in fast collisions. More
difficult is the role of final state correlation. This has
lead to extensive discussion, including the
development of distorted wave Born methods (that
may be incorporated as an extension [1] of Eq. (2),
various discussions of post collision interaction (PCI)
effects [1], and use of various techniques for accurate
evaluation of the final state wavefunction, Ψ−, in
Eq.(2). One such method that has been widely used is
A good deal of data as been taken for double
ionization by photons and by charged particles,
including positrons and anti-protons. The difference
between particles and anti-particles has been directly
attributed to differences in their time evolution due to
time ordering [3]. Since an anti-particle may be
regarded as a particle traveling backward in time, the
time ordering effects are reversed. While some
34
13. Brauner M., Briggs J. S., and Klar H., J. Phys. B 22,
2265 (1989).
observations of excitation-ionization have been done
for electrons and protons, few have been done with
high-energy photons and none have been done with
anti-matter. We expect that excitation-ionization will
probe correlation dynamics somewhat differently than
double ionization.
14. Doerner R., et al., Photonic, Electronc And Atomic
Collisions, Rinton Press, NJ, 2002, p. 27.
15. Corkum P, Phys. Rev. Lett. 71, 1994 (1993).
In summary studies of excitation-ionization are
providing new information as well as new insight into
the dynamics of the few body problem. In particular
progress is being made on the long-standing problem
of electron correlation in the continuum, insight is
being gained on dynamic processes for electron
correlation and new studies are developing on time
correlation between electrons. It is hoped that new
data from synchrotrons will be taken and that it can be
related to data from charged particles, including antimatter.
16. Godunov A. L., and McGuire J. H., J. Phys. B. 34, L223
(2001).
17. Godunov A. L., McGuire J. H., Shakov Kh. Kh.,
Merabet H., Hanni J., Bruch R., and Schipakov V.,
J. Phys. B 34, 5055 (2001).
18. Merabet H., Bruch R., Godunov A. L., McGuire J. H.
and Hanni J., Phys. Rev. A65, 010703(R) (2002).
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