NMR and Quadrupolar Nuclei

Since its development in the 1940s,
nuclear magnetic resonance (NMR) spectroscopy has become one of the most valuable research tools available to chemists. "It's the single most powerful experimental technique for characterizing the chemistry of any kind of sample," says Alan Benesi, "solid, liquid or gaseous."

In the 1980s, researchers added scanning technology that helped take NMR into hospitals, where as magnetic resonance imaging (MRI) it rapidly proved its worth as a diagnostic tool. With less fanfare, the parent technology, NMR spectroscopy, also expanded its usefulness. It is widely used in industry and since the mid-1980s has become important in biomedical research, where it supplements X-ray crystallography as a method for determining the structure of proteins.

Another NMR revolution is now underway, says Benesi, who directs the NMR facility at Pennsylvania State University. New methods will make it feasible to use NMR spectroscopy for a large class of atoms that are in theory observable, but in practice present difficult problems. These atoms, which happen to comprise most of the periodic table, have "quadrupolar" nuclei -- a nonspherical distribution of positive charge in the nucleus -- which makes the NMR spectrum they produce much more complicated than it otherwise would be. "More than 65 percent of the NMR-observable nuclei in the periodic table are quadrupolar," says Benesi, "and most of the time these nuclei are found in solids such as metals, ceramics and minerals."

Benesi's research involves devising mathematics, numerical methods and computer models to simulate the quantum behavior that underlies NMR experiments, and he focuses on quadrupolar nuclei. "I'm trying to understand what's going on with these nuclei, which will help us find ways, new tricks, to get better information from the experiments." Using the Pittsburgh Supercomputing Center's CRAY T3D to great advantage, with unprecedented performance of 39 billion computations a second (Gflops), his simulations have revealed surprising and promising new information.

Bridging the Gap

The ability to get useful information from NMR spectroscopy goes hand-in-hand with computer simulations that link the experiments with theory. "There's an intimate tie between theory and experiment," says Benesi. "For the more unusual experiments especially, you choose a set of parameters and then, basically, you solve the quantum mechanics to predict what you should observe and compare it with what you see. Then you say 'Oops, that's off.' You go back and change the parameters and say 'Oh, that's better,' back and forth between theory and experiment, until you find the best match."

Spinlock: Experiment and Theory
This graph shows signal intensity over time for an NMR "spinlock" experiment (red) with quadrupolar nuclei compared to computer simulation (black) of the experiment. The simulation used Alan Benesi's Series Expansion of Propagators method on the CRAY T3D to calculate the "powder average" of the density matrix. The sample is sodium (23Na) in sodium oxalate.
For quadrupolar nuclei, the mathematics of the theory is complicated enough that the experimental ramifications aren't well understood, and the experiments themselves reveal unusual behavior. A useful technique known as "spinlock," for instance, involves applying a long radio-frequency pulse to the sample to "lock" it at its resonance frequency. For reasons that are not fully understood, explains Benesi, non-quadrupolar nuclei can be spinlocked for much longer than quadrupolar nuclei, where the signal dies away to almost nothing.

With his scientific curiosity piqued, in 1993 Benesi devised his own numerical recipe for simulating NMR. Called Series Expansion of Propagators, the method is especially well suited to the quantum traits of quadrupolar nuclei. During 1995, with help from Penn State colleagues Ken Merz and Jim Vincent and PSC consultant Ravi Subramanya, Benesi implemented his method on the T3D. "These calculations take enormous amounts of computing," says Benesi, "and the T3D makes a huge difference. What used to take me a week (on a Silicon Graphics workstation) now takes about three hours. With this turnaround, the data becomes much more useful to compare against experiments."

The most time-consuming part of Benesi's computations is what's called the "powder average," so called because samples used in solid-state NMR are almost always in powder form. Simulating the NMR signal from such a powder is, basically, a summing up of many individual simulations, each of which corresponds to one among all the possible directions that the huge quantity of individual crystals in the sample could be oriented. The inherent parallelism of this computation is one of the keys to the outstanding performance (76.5 Mflops per processor scales linearly to 39.2 Gflops on 512 processors) Benesi achieves on the T3D.

New Finding: Decay of the Density Matrix

Benesi's ability to compute the powder average led him to a fascinating discovery. What he calculates is a quantum mathematical quantity called the "reduced density matrix," and what he found -- which no one had seen before -- was that in simulations of spinlock with quadrupolar nuclei the reduced powder average density matrix decays sharply to almost zero. Sound familiar? The obvious parallel between density matrix decay in simulations and signal loss in actual experiments suggests a connection between theory and experiment that no one had surmised until Benesi's work.

These results are still being absorbed by researchers at other NMR facilities, and the implications have yet to be sorted out. They come as a surprise because no one anticipated that a powder average density matrix would do what Benesi's computations show. "For an individual crystal," explains Benesi, "the density matrix stays constant, no matter what you do to it. Only when you add the matrices together do you get this behavior. In hindsight, it should have been obvious. There's destructive interference among the different frequencies for the differently oriented crystals."

In recent computations, Benesi arrived at another interesting result: With a second radio-frequency pulse, he can revive the near-death density matrix. "I can make it echo back. I can give a pulse, watch the density matrix decay away, then give a refocussing pulse and make the matrix climb back up, in the same amount of time it took to decay. If we can take advantage of that -- devise a way to keep observable signals from dying, that would be extremely important in solid-state NMR."

Researchers: Alan Benesi, Pennsylvania State University.
Hardware: CRAY T3D
Software: User-developed Code.
Keywords: nuclear magnetic resonance spectroscopy, NMR, magnetic resonance imaging, MRI, quadrupolar nuclei, spinlock, powder average, reduced density matrix, Series Expansion of Propagators.
Related Material on the Web:
Alan J. Benesi, Pennsylvania State University
Alan Benesi's Home Page
Projects in Scientific Computing, PSC's annual research report.
References, Acknowledgements & Credits