We present a model for the Global Quantum Efficiency (GQE) of the MicroBooNE optical units. An optical unit consists of a flat, circular acrylic plate, coated with tetraphenyl butadiene (TPB), positioned near the photocathode of a 20.2-cm diameter photomultiplier tube. The plate converts the ultra-violet scintillation photons from liquid argon into visible-spectrum photons to which the cryogenic phototubes are sensitive. The GQE is the convolution of the efficiency of the plates that convert the 128 nm scintillation light from liquid argon to visible light, the efficiency of the shifted light to reach the photocathode, and the efficiency of the cryogenic photomultiplier tube. We develop a GEANT4-based model of the optical unit, based on first principles, and obtain the range of probable values for the expected number of detected photoelectrons ($N_{rm PE}$) given the known systematic errors on the simulation parameters. We compare results from four measurements of the $N_{rm PE}$ determined using alpha-particle sources placed at two distances from a TPB-coated plate in a liquid argon cryostat test stand. We also directly measured the radial dependence of the quantum efficiency, and find that this has the same shape as predicted by our model. Our model results in a GQE of $0.0055pm0.0009$ for the MicroBooNE optical units. While the information shown here is MicroBooNE specific, the approach to the model and the collection of simulation parameters will be widely applicable to many liquid-argon-based light collection systems.
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