The Density Matrix and its Basic Applications in NMR

Abstract

I hope we now established some qualitative physical concepts and pictures about the fundamentals of NMR after the first three chapters. This chapter will introduce the basics of quantitative analysis of NMR experiments by quantum mechanics. A good background in linear algebra will be very beneficial, even if you have no prior knowledge of quantum mechanics. We will start our discussion with introduction of relevant quantum mechanical concepts and principles necessary for our NMR analysis, with emphasis on the density matrix method. Then density matrix will be applied to explain the concept of coherence, and why normally only -1 coherence is detected in our quadrature detection setup. As an example, the evolution of a system consisting of non-interacting spins will be derived under one of the most famous RF pulse sequences, the Hahn Echo, in the presence of isotropic chemical shift. To account for interactions, we will demonstrate how to break down the dipolar interaction into different coherent components. Subsequently we will introduce the operator formalism, which is based on the density matrix method. The operator formalism will be applied to analyze some basic NMR pulse sequences, including the Stimulated Echo, Solid Echo and adiabatic demagnetization. In this process, the effects of nuclear interaction are manifested by their effect over the evolution of the spin system. In this process, another important concept, the coherence transfer will be demonstrated. It is frequently exploited in various NMR pulse sequences, both in solution and ssNMR. To facilitate the analysis, the transformation into fictitious double and zero quantum space will be introduced. They will be employed to demonstrate the excitation and detection of double quantum coherence by a simple (π2)x−τ−(π2)x sequence. It shares the core principles as other advanced pulse sequences to be discussed in Chapter 6 and 7. To generalize our discussion, the Cogwheel phase cycling will be explained as the optimal strategy for coherence pathway selection. The fictitious spin operators will also be applied to analyze the mechanism of cross polarization (CP), one of the most frequently applied signal enhancement pulse sequence in ssNMR. We will end our discussion with the average Hamiltonian theory (AHT), which will be our main theoretical framework to analyze more complicated pulse sequences in subsequent chapters.