  Case studies of $1T$-TiSe$_2$ and YBa$_2$Cu$_3$O$_{7-\delta}$ have
demonstrated that x-ray diffraction (XRD) studies can be used to trace even
subtle structural phase transitions which are inherently connected with the
onset of superconductivity in these benchmark systems. Yet, the utility of XRD
in the investigation of superconductors like MgB$_2$ lacking an additional
symmetry-breaking structural phase transition is not immediately evident. Even
though, high-resolution powder XRD experiments on MgB$_2$ in combination with
maximum entropy method (MEM) analyses hinted at differences between the
electron density distributions at room temperature and 15K, i.e. below the
$T_c$ of approx. 39K. The high-resolution single-crystal XRD experiments in
combination with multipolar refinements presented here can reproduce these
results, but show that the observed temperature-dependent density changes are
almost entirely due to a decrease of atomic displacement parameters as a
natural consequence of reduced thermal vibration amplitude with decreasing
temperature. Our investigations also shed new light on the presence or absence
of magnesium vacancies in MgB$_2$ samples $-$ a defect type claimed to control
the superconducting properties of the compound. We propose that previous
reports on the tendency of MgB$_2$ to form non-stoichiometric Mg$_{1-x}$B$_2$
phases ($1 - x \sim$0.95) during high-temperature (HT) synthesis might result
from the interpretation of XRD data of insufficient resolution and/or usage of
inflexible refinement models.
