Dynamical Models of Chemical Exchange in Nuclear Magnetic Resonance Spectroscopy

ABSTRACT Chemical exchange line broadening is an important phenomenon in nuclear magnetic resonance (NMR) spectroscopy, in which a nuclear spin experiences more than one magnetic environment as a result of chemical or conformational changes of a molecule. The dynamic process of chemical exchange strongly affects the sensitivity and resolution of NMR experiments and increasingly provides a powerful probe of the interconversion between chemical and conformational states of proteins, nucleic acids, and other biologic macromolecules. A simple and often used theoretic description of chemical exchange in NMR spectroscopy is based on an idealized 2-state jump model (the random phase or telegraph signal). However, chemical exchange can also be represented as a barrier crossing event that can be modeled by using chemical reaction rate theory. The timescale of crossing is determined by the barrier height, the temperature, and the dissipation modeled as collisional or frictional damping. This tutorial explores the connection between the NMR theory of chemical exchange line broadening and strong collision models for chemical kinetics in statistical mechanics. Theoretic modeling and numeric simulation are used to map the rate of barrier crossing dynamics of a particle on a potential energy surface to the chemical exchange relaxation rate constant. By developing explicit models for the exchange dynamics, the tutorial aims to elucidate the underlying dynamical processes that give rise to the rich phenomenology of chemical exchange observed in NMR spectroscopy. Software for generating and analyzing the numeric simulations is provided in the form of Python and Fortran source codes.

A Model of Spectral Line Broadening in Signal Forecasts for Line-intensity Mapping Experiments

Line-intensity mapping observations will find fluctuations of integrated line emission are attenuated by varying degrees at small scales due to the width of the line emission profiles. This attenuation may significantly impact estimates of astrophysical or cosmological quantities derived from measurements. We consider a theoretical treatment of the effect of line broadening on both the clustering and shot-noise components of the power spectrum of a generic line-intensity power spectrum using a halo model. We then consider possible simplifications to allow easier application in analysis, particularly in the context of inferences that require numerous, repeated, fast computations of model line-intensity signals across a large parameter space. For the CO Mapping Array Project and the CO(1–0) line-intensity field at z ∼ 3 serving as our primary case study, we expect a ∼10% attenuation of the spherically averaged power spectrum on average at relevant scales of k ≈ 0.2–0.3 Mpc−1 compared to ∼25% for the interferometric Millimetre-wave Intensity Mapping Experiment targeting shot noise from CO lines at z ∼ 1–5 at scales of k ≳ 1 Mpc−1. We also consider the nature and amplitude of errors introduced by simplified treatments of line broadening and find that while an approximation using a single effective velocity scale is sufficient for spherically averaged power spectra, a more careful treatment is necessary when considering other statistics such as higher multipoles of the anisotropic power spectrum or the voxel intensity distribution.

Multi-Dimensional Nuclear Magnetic Resonance Methods in the Inhomogeneous Magnetic Field

<p>This thesis introduces new NMR techniques which use the inhomogeneous internal magnetic fields present in the pore space of a porous medium exposed to an external magnetic field to obtain information about the pore size and heterogeneities of the the sample. Typically internal field inhomogeneities are regarded as unwanted due to their effect on various material properties such as relaxation and diffusion. However, in the experiments presented here, we choose samples specifically for their inhomogeneous internal fields and use multi-dimensional NMR methods and simulations to obtain our pore space and heterogeneity information. We first describe software developed to specifically simulate the internal magnetic field and diffusion through the pore space of a simple sphere pack system. This software generates a sphere pack and calculates the internal magnetic field generated by z-aligned magnetic dipoles placed at the center of each sphere. The internal magnetic field gradient is also calculated in the pore space. From there, a random walk method is developed and a realistic reflection off a sphere is introduced. We work through the development of this software and the mathematics behind the algorithms used. This simulation is used in all subsequent experimental chapters. We then use a two-dimensional exchange experiment to separate the susceptibility induced line broadening with the broadening caused by diffusion through the inhomogeneous field. We observe off-diagonal line broadening as the mixing time increases. We attempt to quantify this off-diagonal growth by selecting points on either side of the off-diagonal maximum and plotting their average as a function of mixing time. A biexponential fit to the average intensities with respect to mixing time results in a characteristic time and from that a characteristic length as a fraction of bead diameter. This experiment is simulated and a biexponential growth is also observed in the simulated off-diagonal with characteristic lengths comparable to experiment. To obtain a correlation length directly from experiment and not deduce one from a characteristic time, we add a spatial dimension to our exchange experiment in the form of a propagator dimension. This dimension allows us to select 2D spectra based on their Z-displacement. We observe off-diagonal growth due to both an increase in Z-displacement and an increase in mixing time. We move away from the biexponential fit and move to a relationship based on mixing time, effective diffusion, and Z-displacement to directly calculate a characteristic length. We see these same traits in the simulated data which agrees well with experiment. Lastly, we move away from exchange experiments and move to correlating the transverse relaxation time with the internal field offset. We find that there is correlation at large magnetic field offsets and small T2 times which appear to be indicative of sample heterogeneities. To confirm this we use a highly heterogeneous rock core sample which increases the correlations seen at the previous offsets and times. This experiment is more qualitative than the previous two as we do not have a concrete value for the heterogeneity of our samples. The simulation used throughout the thesis, while showing a definite correlation between field offset and T2 relaxation, is unable to accurately simulate the experiment and requires more development.</p>

Multi-Dimensional Nuclear Magnetic Resonance Methods in the Inhomogeneous Magnetic Field

<p>This thesis introduces new NMR techniques which use the inhomogeneous internal magnetic fields present in the pore space of a porous medium exposed to an external magnetic field to obtain information about the pore size and heterogeneities of the the sample. Typically internal field inhomogeneities are regarded as unwanted due to their effect on various material properties such as relaxation and diffusion. However, in the experiments presented here, we choose samples specifically for their inhomogeneous internal fields and use multi-dimensional NMR methods and simulations to obtain our pore space and heterogeneity information. We first describe software developed to specifically simulate the internal magnetic field and diffusion through the pore space of a simple sphere pack system. This software generates a sphere pack and calculates the internal magnetic field generated by z-aligned magnetic dipoles placed at the center of each sphere. The internal magnetic field gradient is also calculated in the pore space. From there, a random walk method is developed and a realistic reflection off a sphere is introduced. We work through the development of this software and the mathematics behind the algorithms used. This simulation is used in all subsequent experimental chapters. We then use a two-dimensional exchange experiment to separate the susceptibility induced line broadening with the broadening caused by diffusion through the inhomogeneous field. We observe off-diagonal line broadening as the mixing time increases. We attempt to quantify this off-diagonal growth by selecting points on either side of the off-diagonal maximum and plotting their average as a function of mixing time. A biexponential fit to the average intensities with respect to mixing time results in a characteristic time and from that a characteristic length as a fraction of bead diameter. This experiment is simulated and a biexponential growth is also observed in the simulated off-diagonal with characteristic lengths comparable to experiment. To obtain a correlation length directly from experiment and not deduce one from a characteristic time, we add a spatial dimension to our exchange experiment in the form of a propagator dimension. This dimension allows us to select 2D spectra based on their Z-displacement. We observe off-diagonal growth due to both an increase in Z-displacement and an increase in mixing time. We move away from the biexponential fit and move to a relationship based on mixing time, effective diffusion, and Z-displacement to directly calculate a characteristic length. We see these same traits in the simulated data which agrees well with experiment. Lastly, we move away from exchange experiments and move to correlating the transverse relaxation time with the internal field offset. We find that there is correlation at large magnetic field offsets and small T2 times which appear to be indicative of sample heterogeneities. To confirm this we use a highly heterogeneous rock core sample which increases the correlations seen at the previous offsets and times. This experiment is more qualitative than the previous two as we do not have a concrete value for the heterogeneity of our samples. The simulation used throughout the thesis, while showing a definite correlation between field offset and T2 relaxation, is unable to accurately simulate the experiment and requires more development.</p>