Dependence of the residual surface resistance of superconducting RF cavities on the cooling dynamics around $T_mathrm{c}$

We report a strong effect of the cooling dynamics through $T_mathrm{c}$ on the amount of trapped external magnetic flux in superconducting niobium cavities. The effect is similar for fine grain and single crystal niobium and all surface treatments including electropolishing with and without 120$^circ$C baking and nitrogen doping. Direct magnetic field measurements on the cavity walls show that the effect stems from changes in the flux trapping efficiency: slow cooling leads to almost complete flux trapping and higher residual resistance while fast cooling leads to the much more efficient flux expulsion and lower residual resistance.