Statistical properties of $^{133}$Xe and the $^{132}$Xe$(n,\gamma)$ cross section
$^{133}$Xe is an interesting case for plasma physics to explore nuclear excitation by electron capture, as the process can be studied using statistical properties of $^{133}$Xe. In this work we present results on $^{133}$Xe from the inverse-Oslo method where we extract the nuclear level density and the $\gamma$-strength function, which is used to calculate the (n,$\gamma$) cross section on $^{132}$Xe. The $\gamma$-strength function of $^{133}$Xe can constrain the estimated decay rate from nuclear excitation by electron capture. The $\mathrm{d}(^{132}\mathrm{Xe},\mathrm{p})^{132}\mathrm{Xe}$ reaction was used to create the compound nucleus $^{133}$Xe, which was recorded with an annular particle telescope and a scintillator array consisting of \la and BGO-shielded HPGe Clover detectors. With the inverse-Oslo method, it is possible to study nuclei that are impossible or unable to manufacture targets from, short lived isotopes, or as in this work, noble gases. We present the extracted nuclear level density, and $\gamma$-strength function for $^{133}$Xe, along with shell-model calculations of the statistical properties of $^{133}$Xe. These are the first statistical properties extracted below 6 MeV for any xenon isotope. We constrain the $^{132}$Xe(n,$\gamma$) $^{133}$Xe cross section and reaction rate using the TALYS reaction code.