  The slow neutron capture ($s$-process) synthesises ~50% of all elements in
the universe heavier than iron, whose abundances are determined by the
competition between neutron capture and nuclear $\beta$-decay rates. The latter
are expected to vary inside hot and dense plasmas such as those found in
$s$-process nucleosynthesis sites. Here, we present a new and general
theoretical study of the effect of local and non-local thermodynamic
equilibrium ((N)LTE) plasmas on $\beta$-decays, using orbital electron capture
(EC) decays in $^{7}$Be as a model case. We begin from the model of Takahashi
and Yokoi to calculate the lepton phase volume of $^{7}$Be as a function of its
ionisation state and excitation level, and consequently, the
configuration-dependent EC decay rate. We then calculate the in-plasma ion
charge state distribution (CSD) and level population distribution (LPD) for a
grid of plasma density and temperatures, using the population kinetics code
FLYCHK. By combining the configuration-dependent EC rate with the CSD and LPD,
we calculate the in-plasma orbital EC rate in $^{7}$Be. The results show a
strong correlation between the half-life and thermodynamic conditions of the
plasma, underlining the importance of measuring decay rates in laboratory
plasmas and the relevance of high precision atomic configuration models. The
model discussed in this work is capable of calculating EC and bound state
$\beta$ decay (BSBD) rates in both low-density NLTE and high-density LTE
plasmas. We conclude by validating our model with state-of-the-art data in
literature on isotopes of Pr and Dy, and by proposing future extension of the
model to laboratory magnetoplasmas and stellar interiors aimed at improving
nucleosynthesis models.
