  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 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.
