  First and second terms of the low-temperature cluster expansion for the free
energy density of a magnetically polarized $XXZ$ spin chain is obtained within
the propagator approach suggested by E. W. Montroll and J. C. Ward. All the
calculations employ only one- and two-magnon infinite-chain spectrums. In the
$XXX$-point the result reproduces the well known S. Katsura's formula obtained
50 years ago by finite-chain calculations.
