An analytical formulation of CPMAS

Abstract

An exact expression for the cross polarization between two spin-1/2 particles is derived from the quantum Liouville equation. This is given in the form of two integrodifferential equations. These can be solved exactly in the static case (no sample spinning) and a powder average easily performed numerically. With magic-angle spinning, the neglect of certain interference terms simplifies the numerical calculation. A further assumption decoupling the calculation of the sidebands gives a very simple formula that is capable of giving a qualitative interpretation of all experimental observations. Examples are given illustrating typical buildup curves and CPMAS matching profiles.

Cross-polarization dynamics in polycrystalline samples

“…The efficacy of polarization transfer varies from site to site and with offset. As a result, relative quantitation is difficult [6][7][8][9][10][11][12][13][14][15][16].…”

“…More detailed theoretical descriptions of the polarization-transfer process have been given [16,22]. While the subject of non-monotonic behavior of the rare-spin magnetization has been raised, in each case the discussion was limited.…”

“…While the subject of non-monotonic behavior of the rare-spin magnetization has been raised, in each case the discussion was limited. For example, to avoid the difficulties of the spin-temperature approach in describing polarization transfer, Marica and Snider [16] derived exact expressions for polarization transfer between an isolated pair of spin-½ particles based on the quantum Liouville equation under both static and MAS conditions, including powder-average results, but without including spin relaxation and spin diffusion in their calculations. The results predict non-monotonic behavior, but the lack of inclusion of spin diffusion limits the practical applicability of their results to isolated spin pairs.…”

Dynamics of Cross Polarization in Solid State Nuclear Magnetic Resonance Experiments of Amorphous and Heterogeneous Natural Organic Substances

“…(1), cannot be used to gain information on the proton relaxation dynamics due to the overestimation of the T i 1r (H) values obtained when thermodynamic approaches are used to describe the CPMAS phenomenon. 27 However, the non-monotonic model applied for humic substances provided a new parameter, b, that is the dipolar coupling between protons and carbons. 20 Since b depends on the angle between the direction of the vector connecting protons to carbons and that of the applied magnetic field (1), we should have expected it to be null at the magic angle used to perform our CP experiments.…”

“…Non-null b values can be explained by the fact that the non-crystalline humic substances do not provide the classical powder-pattern CP spectra, thereby making the dipolar coupling parameter both time and 1-dependent. 27 A residual chemical shift anisotropy is, then, still measured due to the non-ordered distribution of the C AE H vectors along the magic angle.…”