The Hamiltonians of NMR. part I

Abstract

Nuclear magnetic resonance spectroscopy has evolved in complexity to the extent that it is often useful to carry out computer simulations of experiments as a way to refine experimental conditions, to maximize usable information, or to interpret results reliably. The starting point for such simulations is an appropriate quantum mechanical formulation. This article presents many of the physical ideas that are important in high-resolution NMR spectroscopy and then develops expressions for the contribution of each of these to the Hamiltonian for the system under consideration. The Hamiltonians for the Zeeman interaction, scalar coupling, and the chemical shielding effect are discussed with the aim of explaining the quantum mechanics of NMR in a manner understandable to a nontheoretician. Consequently, much of the discussion is explicit and includes points skimmed over in other presentations in the literature. FOREWORDIn this article and in Part 11, we will discuss the Hamiltonians needed to describe highresolution NMR experiments, including those interactions that lead to relaxation. The mathematical expressions for the interactions and their corresponding Hamiltonians become more complicated in the progression from the simple Zeeman term, to dipolar interactions, to chemical shift anisotropy, and so forth. We express all of the interactions in as similar a form as possible, even though doing so involves making some of the simple terms look more complicated than usual. Nevertheless, we believe that using consistent notation simplifies the study of all of the interactions, and thus the reader is not confronted with completely new expressions with the introduction of each new feature of the spin interactions. We encourage readers to persevere through the initial discussion of the format for writing the Hamiltonians, with the expectation that the benefit of consistent notation will outweigh the toil of introducing and becoming familiar with it.We have inserted notes in shaded boxes at numerous places in the discussion; these are not required for understanding the logical flow of the text. The notes either explain relationships to other topics or elaborate on side issues.

Nuclear magnetic resonance studies of quadrupolar nuclei and dipolar field effects

“…Eq. 6.106 is valid in a homogeneous B 0 field; if the evolution frequencies δ I and δ S depend on position, the factor 30 In this case the pitch of the helix of spin S magnetization after the second gradient pulse is an integer submultiple of the wavelength of modulated spin I magnetization.…”

“…Levitt's book [3] also contains a thorough and extremely physical description of these Hamiltonians that is accessible to novices yet useful to advanced spectroscopists. Smith et al [30] have published a good pedagogical review of the Hamiltonians of NMR. These Hamiltonians will 12 The terms "static" and "time dependent" are relative terms that mean "slow" and "fast" compared to the characteristic timescale of the nuclear spin evolution.…”

“…30 In that case the spatially-averaged magnetization density is equal to the m = −n position-independent component of Eq. 6.105:…”

“…The intermolecular dipolar couplings are the source of the distant dipolar field in bulk matter (see §6). The dipolar spin dynamics in the solid state are dominated by couplings between nearby spins (recall the dipolar coupling strength falls off as r −3 ) which overwhelm the macroscopic contribution of distant spins in the multi-spin terms 30 Note that different spins are not coupled to each other by rf fields.…”

“…An n-spin term of ρ N eq remains an n-spin term under any quantum-mechanical evolution if the spins are truly non-interacting, so these terms would never be observable in an NMR experiment if spin couplings were absent. 30 In that case it would be safe to truncate Eq. 1.112 to terms that are at most linear in β:ρ…”