Saturday, November 20, 2010

Simple Scaling and too much to do in too little time

This was a good week. Nathan successfully defended his dissertation with flying colors and our manuscript for Advanced Materials, a high impact journal, was accepted. Our paper is a comment on another paper that previously appeared in Advanced Materials.

One central theme of our work that uses fundamental limits and sum rules to understand the nonlinear-optical response is the idea of scale invariance. The Schrodinger Equation has the property that the shape of the wavefunction does not change when the width of the potential energy function is decreased by a scaling factor b and the depth of the well is simultaneously increased by b squared. Under such a transformation, the wavefunction is compressed by the factor b, but otherwise, the shape remains the same. We call this simple scaling.

The nonlinear-optical quantity of interest to many applications is called the hyperpolarizability. Making the hyperpolarizability as large as possible is an ongoing area of intense research activity. To nobody's surprise, a larger molecule will generally have a larger hyperpolarizability. To better understand what makes a material tick, we have defined a quantity called the intrinsic hyperpolarizability, which is simply the ratio of the hyperpolarizability of a quantum system, divided by the fundamental limit. Interestingly, the intrinsic hyperpolarizability is invariant under simple scaling, so large and small molecules that are related to each other by simple scaling will have the same intrinsic hyperpolarizability.

In examining all the molecules that had been studied for nonlinear-optical applications over a 3 decade period, we found that the large range of hyperpolarizability values could mostly be accounted for by simple scaling. Thus, researchers were making molecules larger and larger but the best intrinsic hyperpolarizabilities remained static at about 0.03 - suggesting that it would be possible to make a factor of 30 improvement; but to get there would undoubtedly requrie a major paradigm shift.

Several years ago, I published a paper that showed how to calculate a related quantity called the intrinsic two-photon absorption (TPA) cross-section. More recently, Javier Perez-Moreno and I published a paper that introduced a rough rule of thumb for determining the intrinsic TPA cross-section - simply divide it by the square of the number of electrons. My earlier paper showed how to determine the number of electrons.

To my delight, my paper on TPA gets lots of citations, not because of what I believe is the beautiful physics of the work, but because scientists refer to my method of counting the number of effective electrons - a quite trivial (and approximate) procedure. To my horror, when comparing molecules, most researchers then go on to divide by the number of electrons rather than by the square of the number of electrons as suggested by Javier's work. This leads to a flawed comparison between molecules.

A recent paper in Advanced Materials reported on TPA cross section measurements of a new class of dendrimers, molecules with ever-branching pieces much like veins and arteries. They referred to my paper when calculating the number of electrons; but, as is usually the case, they divided by the number of electrons and found that the new dendrimer class was an order of magnitude better than the best existing dendrimers - an impressive improvement.

In our comment on this paper, we reanalyzed the data using the N squared rule and found that the new materials were in fact two orders of magnitude better. In addition, the dendrimers within each class, though of vastly differing sizes, all had approximately the same intrinsic TPA cross section. Thus, we were able to show that the authors had made an even more important discovery than they had realized. Usually, comments on a paper point out a negative flaw, leading to strong rebuttals and counter-rebuttals. In this case, all parties were winners.

Unfortunately, these small successes were overshadowed by a pile of work. After the Thanksgiving break, I am going to an NSF meeting in Hawaii (I hate to travel and I hate hot and humid places), where I will be reporting on the results of our projects. I need to prepare a glitzy poster as well as an oral presentation. This, on top of being hopelessly behind in preparing problem sets/solutions, grading, and catching up on lectures for my graduate mechanics class. In addition, I need to write a pile of recommendations and read a 350 page dissertation; the defense will take place early Monday morning.

Smack in the middle of this stressful week, after months of waiting for an estimate, a flooring contractor handed us an estimate and told us that he could get new floors in before Thanksgiving. Things moved fast, requiring us to immediately move large and heavy furniture back and forth between two rooms, which included taking down built-in cabinets and then replacing them, as well painting all the walls. After spending three solid days on manual labor (actually a satisfying break from work), my time pressures have become critical. I cringe at the accumulating piles of manuscripts waiting to be written and the papers that I need to review for journals.

So, how did I handle the stress? I squandered a couple hours writing about my frustrations on this blog. And now, back to work...

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