Clusters can be produced through a supersonic adiabatic expansion of gas with high Hagena parameter into the vacuum through a nozzle, where the collisional mean free path is much smaller than the nozzle outlet size^{10,13}. Atoms or molecules, mediated by the van der Walls force, undergo into the nucleation phase and achieve a quasi-equilibrium state. A cluster ensemble, consisting of 10^{3} to 10^{7} atoms, can be of the size of 10–100′s of nanometers^{7}. The cluster size largely depends on the species of gas, temperature, backing pressure, and the nozzle geometry. The property of cluster can be described by a semi-empirical Hagena scaling law for axisymmetric gas expansion^{19,20}:

$${n}_{c}=a{{Gamma }^{*}}^{b}$$

(1)

where *n*_{c} is the number of particles in a single cluster, *a* and *b* are determined experimentally (Table 1). The semi-empirical Hagena parameter ({Gamma }^{*}) is defined as

$${Gamma }^{*}={k}_{H}frac{{d}^{0.85}{p}_{0}}{{T}^{2.29}}$$

(2)

with *k*_{H}, the gas specific constant (1,650 for argon and 3.85 for helium^{22}), *d,* the orifice diameter in µm, *p*_{0}, the gas backing pressure in bar, and *T*, the gas temperature in Kelvin.

Since this scaling law was obtained from an experiment using sonic nozzles with low backing pressure^{19}, the size of clusters produced from supersonic conical nozzles of a small opening angle at high backing pressure is usually overestimated. The effect of inner boundary layers in the conical nozzle is not taken into account^{23,24}. In the case of a conical nozzle with a jet expansion half-angle of *δ*, *d* should be replaced with the equivalent diameter *d*_{eq} = *0.74d/*tan(*δ*). In the past decades, several experiments were performed to determine the constants *a* and *b* to match the Hagena scaling law in the given interval of *Γ*^{*} (Table 1). As this semi-empirical law was derived without detail consideration of parameters such as the heat from condensation, any additional constraints of flows^{14,19,25}, or boundary layer effects^{23}, the real cluster size and density may largely deviate from the calculated results. Moreover, the scaling law does not give any information about the expansion dynamics as well as the distribution of the gas, which would also affect the cluster growth rate^{26}.

The experimental setup to prove the existence of clusters and to detect the angular distribution of Rayleigh scattered light is shown in Fig. 1a. A CW diode laser (wavelength: 635 nm; power: 3mW) was loosely focused on the gas jet down to 500 µm spot using a lens of 1 m focal length. The polarization axis of the diode laser was selected by a Glan-Taylor polarizer. A gas jet with a valve (Parker Hennifin series 9), assembled with a home-made nozzle extension tube (Fig. 1b,c), puffed argon gas in a vacuum chamber (~ 10^{–3} torr) with the backing pressure ranging from 40 to 80 bar in pulsed mode. The scattered light was collected using an optical fiber (Ocean optics, Φ = 600 µm, NA = 0.22), placed 2 cm away from the gas nozzle. The fiber was mounted on a motorized rotation stage to record the signal at different angles (*θ*) with respect to the laser propagation direction. The scattered signal collected by the fiber was amplified using a photomultiplier tube (Hamamatsu C123497) and read by a digital oscilloscope.

The angular dependence of Rayleigh scattering signal (*S*_{Rayleigh}), if measured in the plane of incident beam polarization, ((widehat{x}widehat{y})-plane, Fig. 1a), can be given as ^{27}:

$${S}_{Rayleigh}propto frac{{I}_{0}}{{lambda }_{L}^{4}}left(1+{cos}^{2}theta right)$$

(3)

where *I*_{0}is the incident beam intensity and *λ*_{L} is the laser wavelength. The angular dependence of the scattered light recorded with a beam, linearly polarized at 45 degree with respect to (widehat{z}) in the (widehat{y}widehat{z})-plane (Fig. 1d), is compared with cos^{2}*θ* function (blue curve). If the polarization axis of the beam is perpendicular to the observation plane, the angular distribution of the scattered light should be uniform. This was experimentally verified with the laser beam polarized along the (widehat{z})-axis (Fig. 1e). All measurements shown in this paper were performed with the laser beam polarized along the (widehat{z})-axis and the scattering signal was taken at θ = 90 deg, unless mentioned otherwise.

A typical temporal evolution of the measured Rayleigh scattering signal is shown in Fig. 2a). After the gas starts to flow into the chamber at t = 0, the valve remains open for different durations varying from 5 to 60 ms. During the valve opening duration larger than 10 ms, the Rayleigh scattering signal is almost constant and disappears within 5 ms of closing the valve. The measurement shows that finite time is needed for the scattered signal to reach the saturation stage. Previous works indicate that the saturation time scale can vary depending on the types of nozzle and solenoid valve used due to the geometrically different expansion of the gas^{10,18}. The saturation phase was not observed for short valve opening time, such as for 5 ms. Therefore, for a stable laser cluster interaction, the valve opening time should be long enough to reach the saturation stage.

The nonlinear increase of Rayleigh scattering signal with backing pressure is useful for determining the onset of clustering according to the relation^{18,26}:

$${S}_{Rayleigh}propto {N}_{0}{n}_{c}propto {N}_{0}{p}^{b}propto {p}^{b+1}propto {p}^{alpha }$$

(4)

where *N*_{0} is the number density of neutral gas, *n*_{c}, the number of atoms in a cluster, *p*, the gas backing pressure. From the known relations, ({N}_{0}propto p) and ({n}_{c}propto {p}^{b}), the dependence of *S*_{Rayleigh} on *p* can be simply denoted as *S*_{Rayleigh}(propto {p}^{alpha }) whereas *b* and *α* are to be determined experimentally. The exponents in Eq. 4 can be derived from the power-law fitting to the scattering signals obtained at different gas backing pressures (Fig. 2b). To best of our knowledge, no previous work has shown the temporal evolution of parameter *α*, but instead, a steady value of *α* was obtained. Figure 2c) visualizes the temporal evolution of *α*, which shows the time-dependence change of cluster characteristics. The parameter *α* at first increases linearly up to 12 ms, then remains constant at the value of 2.8 (pm) 0.2. The error comes from the smoothing of signal by 1,000 points and is calculated to be about less than 10%. In the present experiment, Hagena parameter *Γ*^{*} ranges from 6.2 (times) 10^{4} to 5 (times) 10^{5} in the interval of 10 to 80 bar at room temperature (*T*_{0} = 295.15 K) with the given nozzle geometry (*α* = 6 deg, *d* = 900 µm) as shown in Fig. 1c). In this range of *Γ*^{*}, previous experimental works on cluster size determination^{12,13,14} suggest that the exponent *b* of the Eq. (1) is to be 1.8 (Table 1), giving *α* = 2.8 (Eq. 4), in agreement with our measurement in Fig. 2c). From this relation, one can infer that *n*_{c} turns out to be in a range from 10^{5} to 10^{6} per cluster for the pressure ranging from 10 to 80 bar. Accordingly, the average cluster size (*a*) can be estimated from the number density of a cluster *n*_{c}, as (aapprox 0.1times sqrt[3]{9{n}_{c}}) nm as in ^{16}. In our experimental condition, average size of the cluster is expected to lie in the interval of 10 nm to 40 nm for the given backing pressure range from 10 to 80 bar. As the cluster size (~ 10 nm) is much smaller than the wavelength of diode laser (635 nm), we are indeed in the Rayleigh scattering regime as confirmed from our measurements (Fig. 1d,e).