BERKELEY, CA — For
over half a century, theorists have tried and failed to provide a complete
solution to scattering in a quantum system of three charged particles, one
of the most fundamental phenomena in atomic physics. Such interactions are
everywhere; ionization by electron impact, for example, is responsible for
the glow of fluorescent lights and for the ion beams that engrave silicon
chips.
Now, collaborators at the Department of Energy's Lawrence Berkeley
National Laboratory, Lawrence Livermore National Laboratory, and the
University of California at Davis have used supercomputers to obtain a
complete solution of the ionization of a hydrogen atom by collision with
an electron, the simplest nontrivial example of the problem's last
unsolved component. They report their findings in the 24 December, 1999,
issue of Science magazine.

A REPRESENTATIVE RADIAL WAVE FUNCTION OF TWO ELECTRONS SCATTERED
IN THE COLLISION OF AN ELECTRON WITH A HYDROGEN ATOM

Their breakthrough employs a mathematical transformation of the
Schrödinger wave equation that makes it possible to treat the outgoing
particles not as if their wave functions extend to infinity  as they
must be treated conventionally  but instead as if they simply vanish at
large distances from the nucleus.
"Using this transformation we compute accurate solutions of the
quantummechanical wave function of the outgoing particles, and from these
solutions we extract all the dynamical information of the
interaction," says Bill McCurdy, Berkeley Lab's Associate Laboratory
Director for Computing Sciences and a principal author of the Science
article.
McCurdy and his longtime collaborator Thomas Rescigno, a staff
physicist at Livermore Lab, and their coauthors, doctoral candidate Mark
Baertschy of UC Davis and postdoctoral fellow William Isaacs at Berkeley
Lab, used the SGI/Cray T3E at the National Energy Research Scientific
Computing Center (NERSC) at Berkeley Lab and the IBM Blue Pacific computer
at Livermore Lab for their solution of the threechargedbody scattering
problem.
"An exact firstprinciples solution of the wave function for the
hydrogen atom was vital to establishing the new quantum theory in the
1920s," says Rescigno. "But even today, for systems with three
or more charged particles, no analytic solutions exist"  that is,
there are no explicit solutions to the Schrödinger equation for such
systems.
Rescigno points out that "it wasn't until the late 1950s, using
early computers, that accurate solutions were obtained even for the bound
states of helium," an atom with two electrons closely orbiting the
nucleus. "Scattering problems are a lot more difficult."
As with all scattering problems, the electronionization of a hydrogen
atom begins with a particle incoming at a certain velocity. This electron
interacts with the atom, and two electrons fly out at an angle to each
other, leaving the proton behind. The likelihood that a specific incoming
state will result in an outgoing state with the particles at specific
angles and energies is the "cross section" for that result.
Cross sections of quantummechanical processes are derived from the
system's wave function, solutions of the Schrödinger equation which yield
probabilities of finding the entities involved in a certain state. In
scattering problems, wave functions are not localized but extend over all
space.
Moreover, says McCurdy of the electromagnetic forces between charged
particles, "Coulomb interactions are forever." These infinities
make it impossible to define the final state of scattering exactly.
"The form of the wave function where all three particles are widely
separated is so intractable that no computeraided numerical approach has
been able to incorporate it explicitly."
But, Rescigno notes, "this obviously hasn't stopped people from
working with plasmas and other ionization phenomena. Mathematically,
they've come up with incredibly artful dodges, and some of them even seem
to work."
Earlier this year, however, in the Proceedings of the Royal Society,
Colm T. Whelan of Cambridge University and his colleagues published their
conclusion that all such approximations perform inconsistently and that
those few cases which appear to yield good agreement with experiment
"are largely fortuitous."
By contrast, the method developed by McCurdy and Rescigno and their
coauthors allows the calculation of a highly accurate wave function for
the outgoing state that can be interrogated for details of the incoming
state and interaction in the same way an experimenter would interrogate a
physical system.
They begin with a transformation of the Schrödinger equation called
"exterior complex scaling," invented by Caltech's Barry Simon in
1979 to prove formal theorems in scattering theory. The transformation
leaves the solution unchanged in regions which correspond to physical
reality, producing the correct outgoing waveform based upon the angular
separation and distances of two electrons far from the nucleus.
Once the wave function has been calculated, it must be analyzed by
computing the "quantum mechanical flux," a means of finding the
distribution of probability densities that dates from the 1920s. This
computationally intensive process can yield the probability of producing
electrons at specific energies and directions from the ionized atom.
(Because electrons are identical, there is no way to distinguish between
the initially bound and initially free electron).
The researchers acknowledge important advances made earlier by others
such as Igor Bray and Andres Stelbovics, whose methods could give the
total cross section for ionization of a scattering reaction but could not
give specifics such as the directions or energies of outgoing electrons.
By contrast, says Rescigno, "Our work produces absolute answers at
the ultimate level of detail."
PREDICTED CROSS SECTIONS OF SCATTERED ELECTRONS (SOLID CURVES)
AND EXPERIMENTAL MEASUREMENTS (DOTS) MATCH ALMOST EXACTLY

Comparison with real scattering experiments, such as those recently
published by J. Röder et al, who scattered incoming 17.6 electronvolt
electrons from hydrogen atoms and measured the angles and energies of the
outgoing electrons, prove the accuracy of the new method. The experimental
data points match the graph of the cross sections calculated by Rescigno,
Baertschy, Isaacs, and McCurdy with astonishing exactitude.
"Even if the specific methods have changed, quantum chemistry was
founded when the helium atom with two bound electrons was solved  it
showed that these problems were in principle solvable," McCurdy says.
"What we have done is analogous. The details of our method probably
won't survive, but we've taken a big step toward treating ionizing
collisions of electrons with more complicated atoms and molecules."
"Collisional breakup in a quantum system of three charged
particles," by T.N. Rescigno, M. Baertschy, W.A. Isaacs, and C.W.
McCurdy appears in Science magazine, 24 December 1999. The authors
conclude by noting that the same computing power and tools necessary for
investigating the complexity of increasingly larger systems are also
needed "to answer a basic physics question for one of the simplest
systems imaginable in physics and chemistry."
The Berkeley Lab is a U.S. Department of Energy national laboratory
located in Berkeley, California. It conducts unclassified scientific
research and is managed by the University of California.
