<p>In this study, we concentrate on spin-1/2 systems. A series of tools using the Liouville space method have been developed for simulating of NMR of arbitrary pulse sequences.</p> <p>We have calculated one- and two-spin symbolically, and larger systems numerically of steady states. The one-spin calculations show how SSFP converges to continuous wave NMR. A general formula for two-spin systems has been derived for the creation of double-quantum signals as a function of irradiation strength, coupling constant, and chemical shift difference. The formalism is general and can be extended to more complex spin systems.</p> <p>Estimates of transverse relaxation, R₂, are affected by frequency offset and field inhomogeneity. We find that in the presence of expected B₀ inhomogeneity, off-resonance effects can be removed from R₂ measurements, when ||omega||<= 0.5 gamma\,B₁ in Hahn echo experiments, when ||omega||<=gamma\,B₁ in CPMG experiments with specific phase variations, by fitting exact solutions of the Bloch equations given in the Lagrange form.</p> <p>Approximate solutions of CPMG experiments show the specific phase variations can significantly smooth the dependence of measured intensities on frequency offset in the range of +/- 1/2 gamma\,B₁. The effective R₂ of CPMG experiments when using a phase variation scheme can be expressed as a second-order formula with respect to the ratio of offset to pi-pulse amplitude.</p> <p>Optimization problems using the exact or approximate solution of the Bloch equations are established for designing optimal broadband universal rotation (OBUR) pulses. OBUR pulses are independent of initial magnetization and can be applied to replace any pulse of the same flip angles in a pulse sequence. We demonstrate the process to exactly and efficiently calculate the first- and second-order derivatives with respect to pulses. Using these exact derivatives, a second-order optimization method is employed to design pulses. Experiments and simulations show that OBUR pulses can provide more uniform spectra in the designed offset range and come up with advantages in CPMG experiments.</p> / Doctor of Philosophy (PhD)