  The formation of Population III (Pop III) stars is a critical step in the
evolution of the early universe. To understand how these stars affected their
metal-enriched descendants, the details of how, why and where Pop III formation
takes place needs to be determined. One of the processes that is assumed to
greatly affect the formation of Pop III stars is the presence of a Lyman-Werner
(LW) radiation background, that destroys H$_2$, a necessary coolant in the
creation of Pop III stars. Self-shielding can alleviate the effect the LW
background has on the H$_2$ within haloes. In this work, we perform a
cosmological simulation to study the birthplaces of Pop III stars, using the
adaptive mesh refinement code Enzo. We investigate the distribution of host
halo masses and its relationship to the LW background intensity. Compared to
previous work, haloes form Pop III stars at much lower masses, up to a factor
of a few, due to the inclusion of H$_2$ self-shielding. We see no relationship
between the LW intensity and host halo mass. Most haloes form multiple Pop III
stars, with a median number of four, up to a maximum of 16, at the instance of
Pop III formation. Our results suggest that Pop III star formation may be less
affected by LW radiation feedback than previously thought and that Pop III
multiple systems are common.
