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At the heart of this question is the devilish
difficulty in modeling any system with multiple particles that
interact via the nuclear force. One way to tackle this so-called
many-body problem is to first construct mathematical functions that
describe each particle, and then start multiplying these functions
together to get some idea as to the underlying physics of the system.
It’s a brute force approach that works well enough
to describe light nuclei but is a computational nightmare when it
comes to heavier elements, which have greater numbers of protons and
neutrons.
Describing each particle, which can exist in any
number of quantum states, is complicated enough. But when modeling a
system composed of several dozen or more of these particles, each of
which potentially interacts with every other particle, the
computational complexity quickly becomes astronomical in magnitude.
For example, applying this approach to Nickel-56,
the isotope with 28 protons and 28 neutrons that is the subject of the
research, effectively means solving an equation with 1 billion
variables. The result is "a huge computational effort, which has
become feasible only recently," the authors note.
Chemistry researchers face a similar problem in
studying molecules with many dozens of interacting electrons. Yet all
electron interactions are not created equal, and for decades
scientists have relied on this fact to build streamlined models that
stem the tide of quantum data that needs to be computed.
The key is correlation, the idea that some pairs of
electrons are strongly linked and related. Correlations are what make
it possible to rely on a modicum of data to make predictions about a
complex system like a molecule, an atomic nucleus or even the nightly
crowd at a restaurant.
To decide how to make the best use of a limited
supply of tables, restaurant owners don’t need to think of every
potential customer as likely to interact with every other customer and
move freely from table to table. Rather, most maitre d’s can
anticipate seeing customers who show up in pairs or small groups and
tend to stick together during any given evening. Observing similar
behavior in electrons, quantum chemists have worked for decades to
build and refine the coupled cluster theory, which today has emerged
as the preeminent approach to explaining complex molecular systems.
Correlations also loom large in current theory of
atomic nuclei. For several years, scientists have known that focusing
on the behavior of pairs of nucleons – the generic term for protons
and neutrons – goes a long way to painting an accurate picture of the
entire atomic nucleus. But until now, no one had used coupled-cluster
theory with heavy atomic nuclei.
The researchers first relied on the MSU High
Performance Computing Center and the CMU Center for High Performance
Scientific Computing for the weeks-long task of solving the
billion-variable equation describing Nickel-56, in effect generating a
yardstick by which to measure their more abbreviated model. Next they
compared their energy and wave function data to those generated by the
computationally expensive alternative.
Coupled-cluster theory produced near identical
results and the time spent crunching the numbers – on a standard
laptop – was often measured in minutes or even seconds.
"Sometimes it took longer to input the information
than to run the calculation," said Piotr Piecuch, one of the authors
and a professor in the MSU Department of Chemistry and the MSU
Department of Physics and Astronomy and at National Superconducting
Cyclotron Laboratory.
Other MSU authors include Alex Brown, NSCL
professor; Marta Wloch, a research assistant professor in Chemistry in
Piecuch’s group; and Jeff Gour and Maricris Lodriguito, doctoral
students in Chemistry in Piecuch’s group. The lead author is Mihai
Horoi, associate professor in the Department of Physics at Central
Michigan University. Horoi and Brown did the daunting supercomputing
work of solving the billion-term equation and analyzing the results,
while Gour, Wloch, Lodriguito and Piecuch performed the
coupled-cluster calculations.
Their research bodes well for next-generation
nuclear science. Because of existing and planned accelerators around
the world, the next few decades promise to yield a panoply of heavy
isotopes for study. Theoretical models will need to keep pace with
this expected avalanche of experimental data. To-date, many such
models have treated the nucleus as a relatively undifferentiated
liquid, gas or other set of mathematical averages – all of which tends
to gloss over subtle nuclear nuance.
In contrast, coupled cluster theory may be the only
manageable and scalable model that takes a particle-by-particle
approach.
"We’re really starting to see the nucleus from a
microscopic perspective," said Piecuch. "This gives us a way to start
with particles, in this case nucleons, build an equation and then
solve it – and to do so in a way that is computationally efficient.
Several of the authors are active in the MSU
Mesoscopic Theory Center, a focal point for collaborations between
nuclear and condensed matter physicists, chemists, mathematicians and
scientists from other disciplines. Created in fall 2006, the center
seeks to understand the emergence of complexity from interactions of
elementary constituents. The work of Piecuch and his colleagues
suggests that generic solutions to these mesoscopic problems may be at
hand.
"What’s most exciting may be broad applicability of
the model," Piecuch said. "Coupled cluster theory grew up in chemistry
but seems to work equally well in nuclear physics, where the physical
dimensions and forces are hugely different." |
