# Mo’ Pulses, Mo’ Problems or An Attempt at (Triple Quantum) Coherent Writing

I have spent the last two weeks calibrating and executing multiple pulse sequences. These sequences, once perfected, were used to further study the molecular structure of scandium oxide (Sc2O3). As stated earlier, NMR measures the response of atomic nuclei to radio-frequency pulses. Although a reasonable amount of information is obtainable from single pulses, the multi-pulse sequences I have been working with can be used in order to manipulate a sample’s quantum states. In one particular experiment, the “double frequency sweep,” a four pulse sequence is used to create “triple quantum coherence” in a scandium oxide sample.

In order to understand what is meant by triple quantum coherence, some background information is necessary: In the very small (10-9 meter) world of quantum mechanics, particles can exist in a “superposition of states.” Simple atoms, for instance, typically have a either “spin up” or a “spin down” state. As a result of quantum mechanical weirdness, however, these atoms can exist in both states simultaneously – that is, they exist in a “superposition of states”. The classic illustration of this phenomenon is Schrödinger’s cat, in which a very unlucky cat is placed in a box with a radioactive sample. At a given point of time, the radioactive sample has a 50% chance of emitting a lethal dose of radiation. Quantum mechanically, the radioactive atoms exist in a superposition of states, in which they simultaneously have decayed and not decayed. Therefore, the cat also exists in a superposition of states – it is alive and dead simultaneously. This bizarre state of being can be thought of as “single quantum coherence” – the cat simultaneously exists in two states one energy level apart. However, atoms can exist in more than two states; the scandium nuclei in my experiment, for instance, are spin-7/2, meaning they can have an atomic spin state of ±1/2, ±3/2, ±5/2, or ±7/2 . In triple quantum coherence, atomic nuclei are placed in a superposition of quantum states separated by three energy levels. Schrödinger’s cat, in contrast, only has two quantum states, so it can only exhibit single quantum coherence. However, if the cat were given more quantum states – perhaps it is placed on a tiredness scale ranging from caffeinated (5) through dead (0) – it could exhibit multiple quantum coherence. If the cat were placed in triple quantum coherence, it would simultaneously exist in any two states three levels apart – that is, it could be simultaneously caffeinated (5) and tired (2), just awake (3) and dead (0), or perky (4) and exhausted (1). By using a multiple-pulse sequence, we are effectively bullying Schrödinger’s cat into one of these states and make appropriate measurements.

Establishing triple quantum coherence, however, is no easy task. A superposition of nonadjacent quantum states is a “forbidden transition.” Though not impossible to obtain (despite its name), the existence of such a state is extremely improbable. Additionally, NMR instruments can only detect transitions between adjacent energy levels, so even when triple quantum coherence can be established, it is extremely hard to detect. As a result of these factors, a multiple pulse experiment must be used to both create and measure the triple quantum coherence.

The actual mechanics of the multiple pulse sequence are a bit easier to describe. A total of four pulses are needed in order to generate and read the triple quantum coherence. Firstly, the sample is inserted into the magnetic field at a specific “magic angle” (about 54.7 degrees with respect to the field), and is spun at a rate of 30,000 Hz. This spinning eliminates noise due to the sample’s anisotropies, resulting in a sharper spectrum with narrower lines. The sample is then hit with the first pulse. This hard, high-power burst of radio waves generates the sought-after triple quantum coherence. There is a brief pause after this pulse, during which the triple coherence is able to evolve, after which the sample is hit by a “dual frequency sweep.” This specialized pulse does exactly what it says on the tin, rapidly irradiating the sample with a variety of radio frequencies. The sweep converts the triple quantum coherence (which cannot be observed) into single quantum coherence (which is easily observed) by boosting the population difference between the central transition (that is, between the ½ and -½  spin states). For best results, this sweep is synchronized with the sample’s spinning speed: one sweep occurs every eighth of a rotor period, (that is, once every 4.1666 microseconds). Finally, two softer, lower power pulses are used to “read out” the coherence at the central transition. The four-pulse sequence is then repeated, with the delay between first and second pulses increased by one rotor period. This change in evolution time enables us to correlate the detected single quantum coherence with the unobservable evolution of the triple quantum coherence. After many such repetitions (my experiment required 288 scans, resulting in a total length of about two and a half days), the resulting FIDs are plotted as a function of the time delay between pulses, producing a two-dimensional plot of the results.

In addition to enabling data collection for its own sake, the two dimensional plots produced by these lengthy, double frequency sweep experiments provide a method for analyzing the scandium oxide’s molecular structure and the chemical environment of its scandium nuclei, Specifically, the plots give use data on chemical shifts and quadrupolar coupling. Chemical shifts are the result of electrons orbiting atomic nuclei. Recall that in NMR, specific nuclei resonate at specific frequencies. Electrons surrounding these nuclei, however, slightly change the resonant frequency of the nuclei, producing a chemical shift. These chemical shifts are thus informative of the electron density surrounding the various scandium nuclei in the sample, allowing for a better understanding of scandium oxide’s electronic environment. Additionally, the double frequency sweep provides information about the sample’s quadrupolar coupling. Unlike the spherical atoms often depicted in elementary chemistry books, atomic nuclei are not symmetrical – instead, they have asymmetrical, football shaped charge distributions, which causes them to interact with electric field gradients. This electrical asymmetry and the resulting interactions with electric fields are described by a quadrupolar coupling constant, which can be determined from my experimental data. The data also provides insights into the sample’s bond lengths, coordination numbers, and other structural parameters.

In addition to this data analysis, I am now running other multiple pulse experiments, and will compare the relative efficiencies and effectiveness of these sequences with that of the double frequency sweep. In this way, I hope to determine the most effective method of analyzing triple quantum coherence in scandium nuclei.