  We consider the prospect of using ultracold hydrogen atoms for sympathetic
cooling of fluorine atoms to microkelvin temperatures. We carry out
quantum-mechanical calculations on collisions between cold F and H atoms in
magnetically trappable states and show that the ratio of elastic to inelastic
cross sections remains high across a wide range of temperatures and magnetic
fields. For F atoms initially in the spin-stretched state ($^2$P$_{3/2}$,
$f=m_f=+2$), sympathetic cooling appears likely to succeed from starting
temperatures around 1 K or even higher. This occurs because inelastic
collisions are suppressed by p-wave and d-wave barriers that are 600 mK and 3.2
K high, respectively. In combination with recent results on H + NH and H + OH
collisions [M. L. Gonz\'alez-Mart\'{\i}nez and J. M. Hutson, arXiv:1305.6282
(2013)], this establishes ultracold H atoms as a very promising and versatile
coolant for atoms and molecules that cannot be laser-cooled.
