## 2 Experiment
### Microwave Cavity Resonance Spectroscopy  
The first main diagnostic of this work is microwave cavity resonance spectroscopy (MCRS), which was used to experimentally measure the free electron density in the plasma. This diagnostic technique is based on the determination of the shift in resonance frequency of a standing microwave mode in a cavity due to a change in the permittivity of the medium through which the microwaves propagate. In the past, this measurement has been used to measure the free electron density in a wide variety of plasma types [31; 64; 65; 66]. The difference between the resonance frequency with plasma present \(f_{\mathrm{p}}\) and the vacuum resonance frequency \(f_{1}\) is therefore a direct measure of the volume-averaged free electron density \(\hat{n}_{\mathrm{e}}\). In the current set of experiments, microwaves were introduced using a vector network analyzer (VNA, Keysight E5072A) in the frequency range \(f=5.0\)-\(5.2\,\mathrm{GHz}\), closely around the resonance frequency of the \(\mathrm{TE_{011}}\) mode in the cavity in vacuum (i.e., without plasma), which was determined to be \(f_{1}=5.099\,\mathrm{GHz}\pm 0.150\,\mathrm{MHz}\). This mode was chosen  
Figure 1: Schematic top view of the experimental setup for monitoring dust particle growth in a low-pressure Ar-HMDSO plasma. The plasma was generated in a cylindrical cavity inside a vacuum vessel. The free electron density of the plasma was measured using a vector network analyzer. The \(\mathrm{TE_{011}}\) mode was used for the MCRS measurements, of which the mode pattern is shown in the inset. Simultaneously, a green laser (\(\lambda=532\,\mathrm{nm}\)) was used to perform laser light extinction measurements on the dust cloud. since its mode pattern (a toroidal region with a maximum intensity at \(r=R/2\), see the inset in Figure 1) best resembles the region where the dust particles are expected to be present. The amount of input microwave power into the cavity was fixed at \(P_{\text{microwave}}=$1\,\mathrm{m}\mathrm{W}$\), whereas the amount of reflected power was monitored at a sample rate of \(10\,\mathrm{H}\mathrm{z}\) for the full frequency range. It can be noted that the microwave power is much less than the plasma power, which indicates that the measurements are non-invasive. This resulted in a reflected power spectrum as a function of the microwave frequency \(f\), which represented an (inverted) Lorentzian-type of power response \(P(f)\), of which the frequency corresponding to the minimum in reflected power reveals the resonance frequency at the moment of measurement. The resonance frequency was determined using a Matlab program that fitted the Lorentzian curve to the measured reflected power spectra. In this way, the shift in resonance frequency \(\Delta f=f_{\text{p}}-f_{1}\) has been determined over the course of the evolution of the dusty plasma. The Slater perturbation theorem relates the shift in resonance frequency \(\Delta f\) to the change in permittivity, from which the volume-averaged free electron density \(\hat{n}_{\text{e}}\) in the plasma can be derived as [67]:  
\[\hat{n}_{\text{e}}=\frac{8\pi^{2}\epsilon_{0}m_{\text{e}}f_{\text{p}}^{2}}{e^{ 2}f_{1}}\frac{\Delta f}{\mathcal{V}}, \tag{1}\]  
with the vacuum permittivity denoted by \(\epsilon_{0}\), the plasma frequency by \(f_{\text{p}}\), the elementary charge by \(e\), the vacuum resonance frequency by \(f_{1}\) (i.e., without plasma), and the electric-field-squared-weighted volume ratio by \(\mathcal{V}\).