# Simulation of Nuclear Quadrupole Resonance Spectra of Phenylalanine Using an NV Center Magnetometer

A technical paper about a technique related to MRI: nuclear quadrupole resonance spectroscopy.

# Abstract

The nitrogen vacancy in diamond can be used as a magnetometer to perform nuclear magnetic resonance and nuclear quadrupole resonance. In this study, nuclear quadrupole resonance spectra are simulated for 14N and 2H in deuterated phenylalanine as outlined in Lovchinsky et al. Dependence on applied magnetic field strength and molecule orientation is calculated. Spectra for bulk samples in random orientation are also calculated, leading to Pake doublets. Challenges with experimental implementation of this technique are considered, and countermeasures are suggested. A code base for calculating spectra for any molecules with known quadrupolar coupling constants and asymmetry parameters with a graphical user interface to toggle molecule orientation is provided.

Nuclear spins with $I > \frac{1}{2}$ exhibit a quadrupole moment $Q$ due to an ellipsoidal charge distribution. Due to a local electric field gradient established by a molecule’s configuration, quadrupolar nuclear spins have energy eigenstates determined by the Hamiltonian[1]:

$\hat{H}_q = \frac{e^{2}qQ}{4}[3\hat{S}_z^{2} + \eta(\hat{S}_x^{2} - \hat{S}_y^{2})]$.

The quantity $e^{2}qQ$ is referred to as the quadrupole coupling constant, abbreviated $\overline{Q}$, and $\eta$ denotes the asymmetry parameter. Taking $\hbar$ to be unity such that energy has units of Hz and shifting the eigenenergies by –$\frac{\overline{Q}}{2}$, the energy eigenstates for an $I = 1$ nucleus take the form:

$E_0 = -\frac{\overline{Q}}{2}, E_{\pm1} = \frac{(1 \pm \eta)}{4}\overline{Q}$

Transitions between these eigenstates result in magnetic field noise which can be detected by a nitrogen vacancy (NV) center sensor. These transitions occur at the frequencies:

$\nu_{\pm1} = \frac{3 \pm \eta}{4}\overline{Q}, \nu_0 = \frac{\eta}{2}\overline{Q}$.

It is then straightforward to extract the $\overline{Q}$ and $\eta$ experimentally by taking a zero-field spectrum.

Applying an external DC magnetic field adds an additional Zeeman coupling term to the Hamiltonian:

$\hat{H}_z = -\gamma \vec{S}\cdot \vec{B}$,

where $\gamma$ represents the gyromagnetic ratio of the quadrupolar nucleus of interest. This term leads to changes in the magnetic field noise spectrum that are highly dependent on the strength of $\vec{B}$ and its direction (or equivalently, the orientation of the molecule of interest). These shifts can be approximated as quadratic functions of the components of $\vec{B}$, but in this study they are evaluated numerically. The appeal of nuclear quadrupole resonance (NQR) for single molecules is the possibility of extracting information about orientation in addition to chemical makeup. For each $\overline{Q}$, $\eta$ pair characterizing a nuclear spin variety in a molecule, three spectral lines are expected. Changing the orientation of the magnetic field and its strength would provide their dispersion, which would then be a powerful tool in identifying a molecule in low concentration.

Bulk samples can also be studied with this scheme. Bulk samples represent random orientations of molecules. Their spectra can be predicted by integrating across all possibilities of single molecule orientations. This results in characteristic spectral features known as Pake doublets[2].

# Simulating NQR Spectra for Deuterated Phenylalanine

To simulate NQR spectra, one can use experimentally derived values for $\overline{Q}$ and $\eta$ to calculate eigenenergies numerically. Then the eigenenergy differences can give the expected frequencies of magnetic field noise, which can be plotted as delta functions. To simulate a continuous spectrum, the delta function spectrum can be convolved with a Lorentzian whose FWHM is estimated from the experimental spectral resolution and broadening. In their simulations, Lovchinsky et al. assume linewidths of 5 kHz from a 200 $\mu$s $T_2$ of the dynamically decoupled NV sensor, and add an additional 4 kHz broadening for 14N lines to account for dipolar coupling to 1H spins. These assumptions will be applied in this work, and their validity will be discussed later in the text.

Deuterated phenylalanine spectra are characterized by four classes of quadrupolar nuclei: deuterium in the aromatic group, the C-D2 group, and the C-D bond, as well as the 14N in the amine group (Figure 1). The experimentally determined values of $\overline{Q}$ and $\eta$ for these bonds are listed in Table 1.

##### Table 1

The hydrogens in the hydroxyl group and the amine group need not be considered, as they are exchanged with the environment with high probability and are not deuterated as a result. The gyromagnetic ratios for 2H and 14N are 6.436 MHz/T and 3.077 MHz/T respectively[7].

First the spectra for 14N in phenylalanine were considered. In Figure 2a, the orientation of the phenylalanine molecule is swept over $\theta$ and $\phi$. $\theta$ represents the polar angle between the symmetry axis of the NV center and the C-D axis of the phenylalanine; $\phi$ represents the azimuthal angle. It is clear the spectral lines exhibit a strong dependence on orientation (note that this sweep does not cover the entire two-dimensional angular space of orientations). To experimentally identify the orientation of a phenylalanine molecule, one could sweep over all orientations in simulation and find the closest set of peaks. If the molecule is also unknown, this task could prove quite challenging. Bulk spectra can be obtained by integrating over all angles. In Figure 2b, the characteristic “Pake doublet” spectral shape is obtained[3]. In this study, the Pake doublet is not as symmetrical as the one reported in Lovchinsky et al. Figure 4b, possibly due to Lovchinsky et al.’s use of a 2nd order quadratic approximation for the eigenenergies.

##### Figure 2

The deuterium spectra are comprised of three sets of three frequencies resulting from the transitions in the aromatic group, the C-D2 group, and the C-D bond. The negligible asymmetry parameter for these deuterons results in degeneracy, so only seven peaks per orientation are visible in these spectra. While the 14N spectral features are on the MHz scale, the splittings in the deuterium spectra are on the order of 5 kHz. Figure 3a, like Figure 2a, illustrates a subset of the orientation dependence of the deuterium spectra at 10 mT. In Figure 3b, the spectra for two phenylalanine molecules in different orientations are plotted as a function of field strength. This plot faithfully recreates Figure 4c from Lovchinsky et al. This plot shows the extraordinary complexity that can arise from even a low number of samples. Very fine spectral resolution in experiment would be necessary to make out the features useful for identifying the molecule specimen and orientation. Figure 3c maps the magnetic field strength dependence of the bulk spectrum from deuterium. The inset shows the Pake doublet feature that emerges.

##### Figure 3

These simulations and those from Lovchinsky et al. demonstrate, in principle, that using an NV-center magnetometer for NQR could be a powerful tool for identifying molecules in low concentration and even determining their orientation. However, there are several practical barriers to achieving this. First, deterministically fixing a single molecule within nanometers of the NV center without nearby molecules interfering with the spectroscopy is a major challenge in and of itself. In addition, to justify the low linewidths used in these simulations, Lovchinsky et al. assumed NV center coherence times three times greater than what they achieved experimentally. However, in experiment, they achieved a mean linewidth of 38 kHz for deuterium, far too high to make out the extremely fine features in spectra like those plotted in Figure 3b. Furthermore, even after dynamical decoupling and repetitive readout, the poor signal-to-noise ratios severely obscured spectral features.

Considering the dramatically reduced spectral resolution in experiment, the “pattern-matching” problem that Lovchinsky et al. propose becomes extremely difficult. Deuterium spectral features would be almost completely unrecognizable. The large spacings of the 14N lines would be more recognizable; however, many amino acid amine groups possess extremely similar $\overline{Q}$ and $\eta$s, limiting the usefulness of 14N spectra if amino acids are the subject of the spectroscopy[8].

# Outlook

Subsequent works have taken steps to address the limited resolution achieved by Lovchinsky et al. Provided samples undergo no diffusion, using an external classical clock and compressive sampling methods, it is possible to achieve sub-millihertz resolution when measuring an oscillating magnetic field with an NV center over several hours[9, 10]. With these improved sensing schemes, NQR could serve as a powerful tool for identifying bulk samples of molecules, or single molecules and their orientations. This work’s code base provides a resource for calculating spectra for molecules with known quadrupolar coupling constants and asymmetry parameters and a graphical user interface to toggle molecule orientation. An online database of NQR spectra as a function of orientation and magnetic field strength could allow researchers to search for matching peaks to identify samples. NV center magnetometry for NQR may usher in a new era of high-precision quantum measurement devices which could serve as revolutionary tools in quantum engineering, physiology, and chemistry.

# Acknowledgements

I would like to thank Professor Lee Bassett for his introduction to the subject of NV-center magnetometry and for his feedback on this work.

# Code

The Python script to run these simulations is available here. The code requires the NumPy, Matplotlib, and Astropy libraries.