T1 spin relaxation of muonium in KCl has been studied at low temperatu
res (20 mK to 2 K), where the motion is believed to be band-like, i.e.
the mean free path is longer than the lattice constant. The Celio mod
el, based on the assumption of stochastic hopping of muonium, accurate
ly describes the field dependence of T1 at higher temperatures but fai
ls below 4 K. The measured T1 spin relaxation rates vary weakly with t
emperature below 2 K even though k(B)T at the lowest temperature is we
ll below the estimated muonium bandwidth obtained from the data at hig
her temperatures. This is taken as evidence that muonium is not comple
tely thermalized on the time-scale of the muon lifetime due to the wea
k interaction with phonons at low temperatures.