  We consider in detail the situation of applying a time dependent external
magnetic field to a 87Rb atomic Bose-Einstein condensate held in a harmonic
trap, in order to adiabatically sweep the interatomic interactions across a
Feshbach resonance to produce diatomic molecules. To this end, we introduce a
minimal two-body Hamiltonian depending on just five measurable parameters of a
Feshbach resonance, which accurately determines all low energy binary
scattering observables, in particular, the molecular conversion efficiency of
just two atoms. Based on this description of the microscopic collision
phenomena, we use the many-body theory of T. Koehler and K. Burnett [Phys. Rev.
A 65, 033601 (2002)] to study the efficiency of the association of molecules in
a 87Rb Bose-Einstein condensate during a linear passage of the magnetic field
strength across the 100 mT Feshbach resonance. We explore different,
experimentally accessible, parameter regimes, and compare the predictions of
Landau-Zener, configuration interaction, and two level mean field calculations
with those of the microscopic many-body approach. Our comparative studies
reveal a remarkable insensitivity of the molecular conversion efficiency with
respect to both the details of the microscopic binary collision physics and the
coherent nature of the Bose-Einstein condensed gas, provided that the magnetic
field strength is varied linearly. We provide the reasons for this universality
of the molecular production achieved by linear ramps of the magnetic field
strength, and identify the Landau-Zener coefficient determined by F.H. Mies et
al. [Phys. Rev. A 61, 022721 (2000)] as the main parameter that controls the
efficiency.
