We present measurements of interfacial Gilbert damping due to the spin pumping effect in Ni$_{81}$Fe$_{19}$/W heterostructures. Measurements were compared for heterostructures in which the crystallographic phase of W, either $alpha$(bcc)-W or $beta$(A15)-W, was enriched through deposition conditions and characterized using X-ray diffraction (XRD) and high-resolution cross-sectional transmission electron microscopy (HR-XTEM). Single phase Ni$_{81}$Fe$_{19}$/$alpha$-W heterostructures could be realized, but heterostructures with $beta$-W were realized as mixed $alpha$-$beta$ phase. The spin mixing conductances (SMC) for W at interfaces with Ni$_{81}$Fe$_{19}$ were found to be significantly lower than those for similarly heavy metals such as Pd and Pt, but comparable to those for Ta, and independent of enrichment in the $beta$ phase.
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