December 24, 1999
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.
A representative radial wave function
of two electrons scattered in the collision of an electron with
a hydrogen atom.
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.
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
quantum-mechanical 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
McCurdy and his longtime collaborator Thomas Rescigno, a staff
physicist at Livermore Lab, and their co-authors, 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
three-charged-body scattering problem.
"An exact first-principles 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 electron-ionization 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 quantum-mechanical 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 computer-aided 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
co-authors 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
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 electron-volt
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
"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
"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."
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research and is managed by the University of California. Visit our
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