We report high-resolution measurements of the singular contribution Rb to the thermal boundary resistance between a solid surface and superfluid helium near the superfluid-transition temperature Tλ. The results confirm the observation by Murphy and Meyer that a gap between the cell end and the sidewall leads to an apparent finite-current contribution to Rb. In the absence of such a gap, overall agreement of Rb with theoretical predictions is very good. Remaining small differences require further investigations. Without a sidewall gap and within our resolution we found no finite-current effects over the range 3.9 μW/cm2<Q<221 μW/cm2.
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