Total radiative thermal neutron-capture $gamma$-ray cross sections for the $^{182,183,184,186}$W isotopes were measured using guided neutron beams from the Budapest Research Reactor to induce prompt and delayed $gamma$ rays from elemental and isotopically-enriched tungsten targets. These cross sections were determined from the sum of measured $gamma$-ray cross sections feeding the ground state from low-lying levels below a cutoff energy, E$_{rm crit}$, where the level scheme is completely known, and continuum $gamma$ rays from levels above E$_{rm crit}$, calculated using the Monte Carlo statistical-decay code DICEBOX. The new cross sections determined in this work for the tungsten nuclides are: $sigma_{0}(^{182}{rm W}) = 20.5(14)$ b and $sigma_{11/2^{+}}(^{183}{rm W}^{m}, 5.2 {rm s}) = 0.177(18)$ b; $sigma_{0}(^{183}{rm W}) = 9.37(38)$ b and $sigma_{5^{-}}(^{184}{rm W}^{m}, 8.33 mu{rm s}) = 0.0247(55)$ b; $sigma_{0}(^{184}{rm W}) = 1.43(10)$ b and $sigma_{11/2^{+}}(^{185}{rm W}^{m}, 1.67 {rm min}) = 0.0062(16)$ b; and, $sigma_{0}(^{186}{rm W}) = 33.33(62)$ b and $sigma_{9/2^{+}}(^{187}{rm W}^{m}, 1.38 mu{rm s}) = 0.400(16)$ b. These results are consistent with earlier measurements in the literature. The $^{186}$W cross section was also independently confirmed from an activation measurement, following the decay of $^{187}$W, yielding values for $sigma_{0}(^{186}{rm W})$ that are consistent with our prompt $gamma$-ray measurement. The cross-section measurements were found to be insensitive to choice of level density or photon strength model, and only weakly dependent on E$_{rm crit}$. Total radiative-capture widths calculated with DICEBOX showed much greater model dependence, however, the recommended values could be reproduced with selected model choices. The decay schemes for all tungsten isotopes were improved in these analyses.
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