Proposal of a New Double-Nozzle Technique for In-Gas-Jet Laser Resonance Ionization Spectroscopy

: This paper proposes a new double-nozzle technique for in-gas-jet laser resonance ionization spectroscopy. We explored the functionality of this new technique through detailed gas dynamic and Monte Carlo atom-trajectory simulations, in which results are presented and discussed. The results of similar computer simulations for JetRIS setup (as a typical representative of the conventional in-gas-jet technique nowadays) are also presented and discussed. The direct comparison of calculation results for the proposed new technique with the conventional one shows that the double-nozzle technique has many advantages compared with the one used in the JetRIS setup at GSI for future high-resolution laser spectroscopic study of heaviest elements. To fully implement the proposed new technique in all existing (or under construction) setups for in-gas-jet laser resonance ionization spectroscopy, it will be enough to replace the used supersonic nozzle with the miniature double-nozzle device described in the paper.


Introduction
For the studies of nuclear, atomic and chemical properties of radioactive elements, different techniques using gas-stopping cells are conventionally used. The primary energetic radioactive ions are injected into a gas cell filled with the high-purity buffer gas (usually helium or argon) by passing through a thin entrance window. Due to a number of their collisions with buffer gas atoms, ions decelerate down to the gas flow velocity and are thermalized. To speed up the transport of ions through big-sized gas cells, a DC gradient electric field is applied. In this case, the ion velocity is determined as a product of the applied electric field in [volt cm −1 ] and the ion mobility coefficient in [cm 2 sec −1 volt −1 ].
The extraction of the ions into a high vacuum is finally performed with the use of radiofrequency quadrupoles (RFQ) [7,8,11,[20][21][22][23][24][25][26] or radio-frequency sextupoles (SPIG) [9,15] installed downstream for the ion beam transportation and focusing. Instead of traditional RFQ and SPIG rod structures, simple, compact, and effective gas dynamic RF-only funnels can also be used. The review of the RF-only funnel technique can find in our recent work [27].
The other way to investigate the nuclear and atomic properties of radioactive elements extracted from the gas cell is by using laser spectroscopy techniques. One can read about, e.g., recent reviews [28,29] and presentation reviews of Piet Van Duppen [30].
The most attractive and promising method of laser spectroscopy for studying the properties of radioactive elements is the so-called method of in-gas-jet laser resonance ionization spectroscopy that was first proposed and experimentally tested in KU Leuven [31]. Briefly, it consists of the following. Under the applied DC electric field gradient, the thermalized in these nozzle's profiles, we could not use these two nozzles in our gas dynamic simulations. Contrary, the third nozzle, which shown in the middle of Figure 3 of Ref. [33], and referred to as the "mid-range nozzle", has a simple conical diverging part with a length of 20 mm, throat diameter of 1 mm and exit diameter of 6.6 mm.
The authors of the work [33] note, "The mid-range nozzle seems to be the best overall choice since its resolution and homogeneity are both close to the optimal values found for the other two nozzles". This is an additional reason why we chose this particular nozzle for a detailed computer study of the operation of the JetRIS setup.
The gas dynamic simulation we have performed for the argon stagnation pressure in the gas cell P cell = 100 mbar and background pressure in the vacuum gas-jet chamber is P bg = 6.47 × 10 −3 mbar. The same pressure values were in the JetRIS setup during test fluorescent measurements of neutral 164 Dy atoms inside the supersonic argon jet (see Table  1 and the Figure 4 caption in [33]).

The Exit Flow Channel
The JetRIS gas cell technical drawing is shown in the left part of Figure 5 in [32]. The inner diameter of the gas cell DC-cage is 160 mm, with a length of 164.5 mm. The total length of the DC funnel (five electrodes having gradually decreasing diameters) placed behind the DC cage is 63.5 mm. The exit flow channel between the DC funnel and the nozzle has a diameter of 12 mm and a length of 34.6 mm. Due to the large diameter of the DC-cage (160 mm) and only 1 mm of the nozzle throat diameter, the average velocity of the gas flow inside the DC-cage is only about 8 mm/s. That is why it is necessary to use the DC gradient field inside the cell to speed up the transport of the decelerated radioactive ions.
Early, we were making detailed (gas dynamic + Monte Carlo) simulations for different big-sized gas-stopping cells [6,[17][18][19]. E.g., the results of our simulations for the UNICELL project [51], which is also under development at GSI, presented in [17]. But here, we should notice that simulations of the 164 Dy ions transport through the main body of the JetRIS gas cell having an inner diameter of 160 mm to the hot filament installed inside the exit flow channel have been carefully done by the authors of [32]. Therefore, it does not make much sense to repeat it, and we decided to start our gas dynamic simulation from the entrance to this exit flow channel (12 mm in diameter), where the average gas velocity is a factor of 178 higher than it is inside the DC-cage region. Notice that the evaporated from the surface of hot filament neutral atoms are moving through the exit flow channel and then through the nozzle into the vacuum gas-jet chamber only under the effect of the gas flow.
Monte Carlo atom-trajectory simulations began from the filament tip, where neutral 164 Dy atoms first appear in the gas flow after evaporation. From the technical drawing presented in [32] we determined that the filament in the JetRIS setup has been installed at about 22.5 mm from the nozzle.
Results of the gas dynamic simulation for the gas velocity flow field inside the exit flow channel are shown in Figure 1. Figure 2 shows the radial gas velocity profile in the middle of this exit flow channel. The parabolic gas velocity profile, typical for the laminar viscous gas flow inside a tube, is visible. The corresponding Reynolds number is equal to 123.
Calculation results for the time-of-flight distribution 164 Dy atoms from the filament tip to the nozzle throat are shown in Figure 3. The average atoms' extraction time is T = 7.0 ms, and the time spread determined as the full width at half maximum (FWHM) is ∆T = 1.4 ms. The corresponding extraction efficiency of 164 Dy atoms from the exit flow channel is 94.5 ± 2.7%.
2.2. GSI Nozzle at Gas Cell Pressure P cell = 100 mbar and P bg = 6.47 × 10 −2 mbar Results of the gas dynamic simulation for argon gas density, temperature, velocity, and Mach number flow fields for the GSI nozzle are shown in Figure 4.        Calculation results for the time-of-flight distribution 164 Dy atoms from the filame tip to the nozzle throat are shown in Figure 3. The average atoms' extraction time is T 7.0 ms, and the time spread determined as the full width at half maximum (FWHM) is = 1.4 ms. The corresponding extraction efficiency of 164 Dy atoms from the exit flow chan is 94.5 ± 2.7%.      We have used the results of the gas dynamic simulation shown in Figure 4 for Monte Carlo simulations of the 164 Dy atomic beam. These results are presented in the next four Figures. The calculated fraction of the extracted from the gas cell 164 Dy atomic beam inside the region of 10 mm (the Laser-1 diameter in [33]) as a function of distance from the GSI nozzle exit is shown in Figure 5. The decrease of the dysprosium atomic beam fraction with the distance from the GSI nozzle is simply explained by the gas jet divergence. The atomic beam diffusion losses inside the GSI nozzle are equal to 15.6 ± 0.6% and are considered in the data presented in Figure 5. Figure 6 demonstrates how the calculated longitudinal velocity spread of the dysprosium atomic beam depends on the distance from the GSI nozzle. The data averaged in the radial plane over the Laser-1 beam diameter makes it possible to compare our calculations directly and simply with future experiments at the JetRIS setup. Moreover, such calculations allow for getting crucial quantitative information on the quality of the cooled in the supersonic gas jet atomic beams. We have used the results of the gas dynamic simulation shown in Figure 4 for Monte Carlo simulations of the 164 Dy atomic beam. These results are presented in the next four Figures. The calculated fraction of the extracted from the gas cell 164 Dy atomic beam inside the region of 10 mm (the Laser-1 diameter in [33]) as a function of distance from the GSI nozzle exit is shown in Figure 5. The decrease of the dysprosium atomic beam fraction with the distance from the GSI nozzle is simply explained by the gas jet divergence. The atomic beam diffusion losses inside the GSI nozzle are equal to 15.6 ± 0.6% and are considered in the data presented in Figure 5.   Figure 6 demonstrates how the calculated longitudinal velocity spread of the dysprosium atomic beam depends on the distance from the GSI nozzle. The data averaged in the radial plane over the Laser-1 beam diameter makes it possible to compare our calculations directly and simply with future experiments at the JetRIS setup. Moreover, such calculations allow for getting crucial quantitative information on the quality of the cooled in the supersonic gas jet atomic beams.

GSI Nozzle at Gas Cell Pressure Pcell = 50 mbar and Pbg = 3.23 × 10 −2 mbar
To check how the supersonic gas jet structure and parameters extracted into the jet 164 Dy atomic beam will look at lower stagnation gas cell pressure, we performed calculations, which are similar to the calculations described above in Section 2.2 for the same pumping capacity, but for the gas cell pressure of Pcell = 50 mbar. The results of the gas dynamic simulation for argon gas density, temperature, velocity, and Mach number flow fields for the GSI nozzle at Pcell = 50 mbar are shown in Figure 9. 2.3. GSI Nozzle at Gas Cell Pressure P cell = 50 mbar and P bg = 3.23 × 10 −2 mbar To check how the supersonic gas jet structure and parameters extracted into the jet 164 Dy atomic beam will look at lower stagnation gas cell pressure, we performed calculations, which are similar to the calculations described above in Section 2.2 for the same pumping capacity, but for the gas cell pressure of P cell = 50 mbar. The results of the gas dynamic simulation for argon gas density, temperature, velocity, and Mach number flow fields for the GSI nozzle at P cell = 50 mbar are shown in Figure 9. Results of the Monte Carlo simulations for the case of 50 mbar gas cell pressure are shown in Figures 10 and 11. In addition, we have included the results of calculations presented above in Figures 5 and 6 for the case of Pcell = 100 mbar for comparison. Notice that the atomic beam diffusion losses inside the GSI nozzle at Pcell = 50 mbar are equal to 27.7 ± 0.8%, about factor 2 higher than that at Pcell = 100 mbar. Results of the Monte Carlo simulations for the case of 50 mbar gas cell pressure are shown in Figures 10 and 11. In addition, we have included the results of calculations presented above in Figures 5 and 6 for the case of P cell = 100 mbar for comparison. Notice that the atomic beam diffusion losses inside the GSI nozzle at P cell = 50 mbar are equal to 27.7 ± 0.8%, about factor 2 higher than that at P cell = 100 mbar. Figure 9. Results of the gas dynamic simulation: (a) density, (b) temperature, (c) velocity and (d) Mach number flow fields for GSI nozzle. The stagnation pressure and temperature in the gas cell are Pcell = 50 mbar and Tcell = 296 K, correspondingly. Background pressure in the gas-jet chamber is Pbg = 3.23 × 10 −3 mbar. Black arrowed lines show the gas flow directions.
Results of the Monte Carlo simulations for the case of 50 mbar gas cell pressure are shown in Figures 10 and 11. In addition, we have included the results of calculations presented above in Figures 5 and 6 for the case of Pcell = 100 mbar for comparison. Notice that the atomic beam diffusion losses inside the GSI nozzle at Pcell = 50 mbar are equal to 27.7 ± 0.8%, about factor 2 higher than that at Pcell = 100 mbar.

Historical Notes
The new double-nozzle technique for in-gas-jet resonance ionization laser spectroscopy we propose here is based on gas dynamic cooling of low-energy molecular and ion beams. This idea was proposed for the first time in 1979 [52,53] and described also, e.g., in [5,39,[54][55][56][57][58][59][60][61]. It consists of direct injection of primary beams into supersonic carrier gas jet using a specially designed supersonic nozzle. The converging-diverging supersonic nozzle has a narrow inner tube on its axis which passes through a gas stagnation chamber into a region of supersonic carrier gas expansion (see Figure 12). Then, this inner tube particle of interest is injected into the expanded supersonic gas jet. A lowpressure zone forms immediately after the tube, so this supersonic carrier gas jet acts as a

Historical Notes
The new double-nozzle technique for in-gas-jet resonance ionization laser spectroscopy we propose here is based on gas dynamic cooling of low-energy molecular and ion beams. This idea was proposed for the first time in 1979 [52,53] and described also, e.g., in [5,39,[54][55][56][57][58][59][60][61]. It consists of direct injection of primary beams into supersonic carrier gas jet using a specially designed supersonic nozzle. The converging-diverging supersonic nozzle has a narrow Atoms 2023, 11, 88 9 of 22 inner tube on its axis which passes through a gas stagnation chamber into a region of supersonic carrier gas expansion (see Figure 12). Then, this inner tube particle of interest is injected into the expanded supersonic gas jet. A low-pressure zone forms immediately after the tube, so this supersonic carrier gas jet acts as a vapor-jet pump. It provides for the possibility of effective differential pumping in the case of ion beam injection [60]. and ion beams. This idea was proposed for the first time in 1979 [52,53] a also, e.g., in [5,39,[54][55][56][57][58][59][60][61]. It consists of direct injection of primary beams int carrier gas jet using a specially designed supersonic nozzle. The convergin supersonic nozzle has a narrow inner tube on its axis which passes th stagnation chamber into a region of supersonic carrier gas expansion (see Fig this inner tube particle of interest is injected into the expanded supersonic g pressure zone forms immediately after the tube, so this supersonic carrier ga vapor-jet pump. It provides for the possibility of effective differential pumpi of ion beam injection [60]. Figure 12. Schematic of the method of gas dynamic cooling of low-energy molecular beams. Picture from Refs. [39,61]. See the text for details. Tо show the feasibility and effectivity of the described gas dyna experimentally. A gas dynamic molecular beam source setup was constr middle of the 1980s at the Leningrad Nuclear Physics Institute (LNPI) in Gat As a result of experimental investigations, it has been obtained the effective c Figure 12. Schematic of the method of gas dynamic cooling of low-energy molecular and ion beams. Picture from Refs. [39,61]. See the text for details.
To show the feasibility and effectivity of the described gas dynamic method experimentally. A gas dynamic molecular beam source setup was constructed in the middle of the 1980s at the Leningrad Nuclear Physics Institute (LNPI) in Gatchina [56,57]. As a result of experimental investigations, it has been obtained the effective cooling of hot lead iodine (PbI) molecular beam in the supersonic nitrogen jet from 900 K down to 20 K was an essential part of our PhD work [54]. Notice that the two-atomic molecules PbI are free radicals produced via the thermal decomposition of three-atomic molecules PbI 2 inside the separately heated thin quartz capillary installed into the inner tube of the supersonic nozzle shown in Figure 12.
The description and schematic layout with photos of some parts of this gas dynamic molecular beam source setup in LNPI one can find in our presentation at Institute for Medium Energy Physics (IMEP) (renamed after 2004 to "Stefan Meyer Institute" (SMI)), Vienna [61].

Double-Nozzle Design
To implement the proposed here double-nozzle technique for in-gas-jet laser resonance ionization spectroscopy, it is sufficient to connect the short conical convergent-divergent nozzle of the gas cell, which we will call the 1st nozzle, with the 2nd nozzle, which designs schematically shown in Figure 12.
Schematic views of the double-nozzle design combined with results of the gas dynamic simulation for argon temperature and velocity flow fields are shown in Figures 13 and 14, correspondingly.
A gas vortex that one can see in Figures 13 and 14 appears in the region between the 1st nozzle throat and the exit of the inner tube in the 2nd nozzle due to the viscous gas jet interaction with the walls. This vortex around the supersonic jet flowing out of the gas cell protects the dysprosium atoms from hitting the walls, thus reducing the diffusion losses of the neutral atomic beam extracted from the gas cell. The result of the Monte Carlo calculation shows that only 9.5 ± 0.4% of the removed from the gas cell atomic beam is lost inside this double-nozzle device.
A gas vortex that one can see in Figures 13 and 14 appears in the region between the 1st nozzle throat and the exit of the inner tube in the 2nd nozzle due to the viscous gas jet interaction with the walls. This vortex around the supersonic jet flowing out of the gas cell protects the dysprosium atoms from hitting the walls, thus reducing the diffusion losses of the neutral atomic beam extracted from the gas cell. The result of the Monte Carlo calculation shows that only 9.5 ± 0.4% of the removed from the gas cell atomic beam is lost inside this double-nozzle device.  The main design parameters of the double-nozzle device used in simulations are listed in Table 1.  The main design parameters of the double-nozzle device used in simulations are listed in Table 1. 3.3. Double-Nozzle at P cell = 100 mbar, P noz = 81 mbar and P bg = 2 × 10 −2 mbar Results of the gas dynamic simulation for argon gas density, temperature, velocity, and Mach number flow fields for the double-nozzle are shown in Figure 15.

Effect of the Pumping Speed
To investigate the effect of the pumping speed on the 164 Dy atomic beam parameters in the supersonic gas jet at fixed stagnation pressures Pcell = 100 mbar, Pnoz = 81 mbar, we have made corresponding gas dynamic and Monte Carlo simulations.
In addition to the previously considered variant of 1300 l/s pumping speed (it corresponds to Pbg = 2 × 10 −2 mbar), we have performed calculations for the pumping speed of 2600 l/s (Pbg = 1 × 10 −2 mbar) and 650 l/s (Pbg = 4 × 10 −2 mbar). The results of these calculations are presented in following

Effect of the Pumping Speed
To investigate the effect of the pumping speed on the 164 Dy atomic beam parameters in the supersonic gas jet at fixed stagnation pressures P cell = 100 mbar, P noz = 81 mbar, we have made corresponding gas dynamic and Monte Carlo simulations.
In addition to the previously considered variant of 1300 l/s pumping speed (it corresponds to P bg = 2 × 10 −2 mbar), we have performed calculations for the pumping speed of 2600 l/s (P bg = 1 × 10 −2 mbar) and 650 l/s (P bg = 4 × 10 −2 mbar). The results of these calculations are presented in following Figures 20-22. To provide a simple and direct comparison of the double-nozzle technique with the case of the GSI nozzle, we have included the results of calculations for the GSI setup [32,33] presented above in Section 2.

Effect of the Input (Stagnation) Pressure in the 2nd Nozzle
To make sure that the proposed here double-nozzle technique works effectively in a wide range of input pressures in the 2nd nozzle, we performed, in addition, gas dynamic and Monte Carlo simulations for two other pressures Pnoz and at fixed gas cell pressure Pcell = 100 mbar and pumping capacity of 1300 l/s. These variants are Pnoz = 162 mbar (the corresponding Pbg = 3 × 10 −2 mbar) and Pnoz = 263 mbar (the corresponding Pbg = 4 × 10 −2 mbar).
The results of these Monte Carlo calculations are presented in Figures 23 and 24. The corresponding results for the case of the GSI nozzle we added for illustration and comparison.

Effect of the Input (Stagnation) Pressure in the 2nd Nozzle
To make sure that the proposed here double-nozzle technique works effectively in a wide range of input pressures in the 2nd nozzle, we performed, in addition, gas dynamic and Monte Carlo simulations for two other pressures P noz and at fixed gas cell pressure P cell = 100 mbar and pumping capacity of 1300 l/s. These variants are P noz = 162 mbar (the corresponding P bg = 3 × 10 −2 mbar) and P noz = 263 mbar (the corresponding P bg = 4 × 10 −2 mbar).
The results of these Monte Carlo calculations are presented in Figures 23 and 24. The corresponding results for the case of the GSI nozzle we added for illustration and comparison. and Monte Carlo simulations for two other pressures Pnoz and at fixed gas cell pressure Pcell = 100 mbar and pumping capacity of 1300 l/s. These variants are Pnoz = 162 mbar (the corresponding Pbg = 3 × 10 −2 mbar) and Pnoz = 263 mbar (the corresponding Pbg = 4 × 10 −2 mbar).
The results of these Monte Carlo calculations are presented in Figures 23 and 24. The corresponding results for the case of the GSI nozzle we added for illustration and comparison.

Effect of the Stagnation Pressure in the Gas Cell
To see how the double-nozzle technique will work at twice less stagnation pressure in the gas cell, we performed gas dynamic and Monte Carlo simulation for the case of Pcell = 50 mbar and input gas pressure Pnoz = 81 mbar. The pumping capacity of 1300 l/s corresponds to Pbg = 1.5 × 10 −2 mbar.
The Monte Carlo calculation results are presented in Figures 25 and 26. The corresponding results for the previously considered case of stagnation Pcell = 100 mbar and Pbg = 2 × 10 −2 mbar we added for illustration and comparison.

Effect of the Stagnation Pressure in the Gas Cell
To see how the double-nozzle technique will work at twice less stagnation pressure in the gas cell, we performed gas dynamic and Monte Carlo simulation for the case of P cell = 50 mbar and input gas pressure P noz = 81 mbar. The pumping capacity of 1300 l/s corresponds to P bg = 1.5 × 10 −2 mbar.
The Monte Carlo calculation results are presented in Figures 25 and 26. The corresponding results for the previously considered case of stagnation P cell = 100 mbar and P bg = 2 × 10 −2 mbar we added for illustration and comparison.
To see how the double-nozzle technique will work at twice less stagnation pressure in the gas cell, we performed gas dynamic and Monte Carlo simulation for the case of Pcell = 50 mbar and input gas pressure Pnoz = 81 mbar. The pumping capacity of 1300 l/s corresponds to Pbg = 1.5 × 10 −2 mbar.
The Monte Carlo calculation results are presented in Figures 25 and 26. The corresponding results for the previously considered case of stagnation Pcell = 100 mbar and Pbg = 2 × 10 −2 mbar we added for illustration and comparison.  Notice that the calculated diffusion losses of the atomic beam inside the doublenozzle device at Pcell = 50 mbar is 18.3 ± 0.6%, a factor of 2 higher than that for Pcell = 100 mbar, and it looks reasonable. Nevertheless, the data for two different Pcell values look very similar in Figures 25 and 26. For comparison, it is not so for the GSI nozzle (see Figures 10 and 11).

Effect of the Laser-1 Beam Diameter
When the distance between adjacent rods of the curved RFQ (the curved RFQ allows for the axial Laser-1 beam injection into the gas jet) is less than considered above the laser beam diameter of 10 mm, one should use the Laser-1 beam of a smaller diameter.
It is essential to know how the JetRIS (or any other similar project) overall efficiency and quality of the extracted into gas jet atomic beam depend on the Laser  Notice that the calculated diffusion losses of the atomic beam inside the double-nozzle device at P cell = 50 mbar is 18.3 ± 0.6%, a factor of 2 higher than that for P cell = 100 mbar, and it looks reasonable. Nevertheless, the data for two different P cell values look very similar in Figures 25 and 26. For comparison, it is not so for the GSI nozzle (see Figures 10 and 11).

Effect of the Laser-1 Beam Diameter
When the distance between adjacent rods of the curved RFQ (the curved RFQ allows for the axial Laser-1 beam injection into the gas jet) is less than considered above the laser beam diameter of 10 mm, one should use the Laser-1 beam of a smaller diameter.
It is essential to know how the JetRIS (or any other similar project) overall efficiency and quality of the extracted into gas jet atomic beam depend on the Laser-1 beam diameter. Some information about one can get from the results of our Monte Carlo simulations shown in Figures 27 and 28. for the axial Laser-1 beam injection into the gas jet) is less than considered above the laser beam diameter of 10 mm, one should use the Laser-1 beam of a smaller diameter.
It is essential to know how the JetRIS (or any other similar project) overall efficiency and quality of the extracted into gas jet atomic beam depend on the Laser-1 beam diameter. Some information about one can get from the results of our Monte Carlo simulations shown in Figures 27 and 28.

Monte Carlo Simulations for Nobelium Atoms in the Gas Jet
The goal of the JetRIS project [32,33] under development at GSI is the study of the nuclear properties of nobelium isotopes with high precision by laser spectroscopy in an argon supersonic gas jet. That is why we performed, in addition, a Monte Carlo simulation for 253 No isotope, as well. For the Monte Carlo simulation of the nobelium atomic beam in a supersonic gas jet, we used the same results of gas dynamic simulation (see Figure 15) for the 164

Monte Carlo Simulations for Nobelium Atoms in the Gas Jet
The goal of the JetRIS project [32,33] under development at GSI is the study of the nuclear properties of nobelium isotopes with high precision by laser spectroscopy in an argon supersonic gas jet. That is why we performed, in addition, a Monte Carlo simulation for 253 No isotope, as well. For the Monte Carlo simulation of the nobelium atomic beam in a supersonic gas jet, we used the same results of gas dynamic simulation (see Figure 15) for the 164 Dy atomic beam. These results are shown in Figures 29-31. Results for the 164 Dy we added for comparison.
As shown in Figure 29, the decrease of the beam fractions with the distance from the 2nd nozzle exit is near the same for both dysprosium and nobelium atomic beams. It reflects that thermalized argon dysprosium and nobelium atoms equally follow the supersonic jet expansion. At the same time, the longitudinal velocity spreads shown in Figure 30 are different. It explains by the mass difference of 164 Dy and 253 No atoms because the velocity spread of the atomic beam thermalized in the gas should be inversely proportional to the square root of the atom mass. Figure 31 shows the 253 No atomic beam longitudinal velocity distributions for different distances from the 2nd nozzle exit for illustration. velocity spread (FWHM) for different Laser-1 beam diameters as a function of the distance from the 2nd nozzle exit. The data averaged in the radial plane for each Laser-1 beam diameter. Number of calculated atoms is 10,000 for each variant.

Monte Carlo Simulations for Nobelium Atoms in the Gas Jet
The goal of the JetRIS project [32,33] under development at GSI is the study of the nuclear properties of nobelium isotopes with high precision by laser spectroscopy in an argon supersonic gas jet. That is why we performed, in addition, a Monte Carlo simulation for 253 No isotope, as well. For the Monte Carlo simulation of the nobelium atomic beam in a supersonic gas jet, we used the same results of gas dynamic simulation (see Figure 15) for the 164 Dy atomic beam. These results are shown in Figures 29-31. Results for the 164 Dy we added for comparison.   253 No atomic beam inside the gas jet of 10 mm in diameter (it is the Laser-1 diameter) for the gas cell pressure P cell = 100 mbar, P noz = 81 mbar and P bg = 2 × 10 −2 mbar as a function of distance from the 2nd nozzle exit. Number of calculated atoms is 10,000 for each case.     As shown in Figure 29, the decrease of the beam fractions with the distance from the 2nd nozzle exit is near the same for both dysprosium and nobelium atomic beams. It reflects that thermalized argon dysprosium and nobelium atoms equally follow the supersonic jet expansion. At the same time, the longitudinal velocity spreads shown in Figure 30 are different. It explains by the mass difference of 164 Dy and 253 No atoms because the velocity spread of the atomic beam thermalized in the gas should be inversely proportional to the square root of the atom mass.

Discussion and Outlook
In Section 2 of the article, we have investigated by means of detailed computer experiments the conventional nowadays technique of the so-called in-gas-jet laser resonance ionization spectroscopy. As a typical representative of this technique, we have chosen for the numerical computer analysis the setup of the JetRIS project [32,33]. The JetRIS project is under development now at GSI, Darmstadt.
Results of the gas dynamic simulation for the exit flow channel of 12 mm in diameter are presented in Figures 1 and 2. It is a laminar gas flow with a Reynolds number of 123. The calculated Monte Carlo simulation time-of-flight of evaporated 164Dy neutral atoms from the hot filament tip to the nozzle throat is 7.0 ± 1.4 ms (see Figure 3), the extraction efficiency is 94.5 ± 2.7%, or in other words, the atomic beam diffusion losses are equal to 5.5 ± 0.3%. The gas dynamic + Monte Carlo simulations show that the additional diffusion losses of the 164Dy atomic beam inside the GSI nozzle at P cell = 100 mbar are equal to 15.6 ± 0.6%. The graphic description of the calculated argon supersonic jet (flow fields of gas density, velocity, temperature, and Mach number) flowing out of the GSI nozzle into the gas-jet chamber is shown in Figures 4 and 9 for the gas cell pressure of 100 mbar and 50 mbar, correspondingly. Here the strong effect of the gas viscosity of the supersonic jet structure and parameters is visible.
Notice that the 164Dy atomic beam diffusion losses inside the GSI nozzle for P cell = 50 mbar 27.7 ± 0.8% are considerably higher than that at P cell = 100 mbar, which looks reasonable. Moreover, the production efficiency of excited by the Laser-1 dysprosium atoms in the gas jet is also less in the case of P cell = 50 mbar (see Figure 10). As a result, the atomic beam quality, shown in Figure 11 as a longitudinal velocity spread, is also worse. Section 3 of the article describes and explores our proposal for a new double-nozzle technique. By means of gas dynamics and Monte Carlo simulations, we have shown that this new technique has many advantages over the conventional technique described in Section 2 of the article. E.g., the atomic beam diffusion losses inside the double-nozzle system are equal only to 9.5 ± 0.4%, which is in factor 1.6 less than in the case of the use of GSI nozzle at the same gas cell pressure (P cell = 100 mbar) and having the same nozzle throat diameter of 1.0 mm. The double nozzle method is much more efficient (see the direct comparison in Figures 16, 21 and 23), and it also allows obtaining better atomic beam quality (expressed in longitudinal velocity spread) for a longer downstream distance from the 2nd nozzle exit (see the direct comparison in Figures 17, 22 and 24).
The proposed double-nozzle technique can effectively work in various P cell and Pnoz pressures. Therefore, there is no need here for the use of any butterfly valve placed in front of the vacuum pump, as it is in [32] to adjust the background pressure in the gas-jet chamber under changing the P cell or using the nozzles having different throat diameters and profiles of its diverging supersonic part. The butterfly valve allows for decreasing the nominal capacity of the used vacuum pump, but it cannot increase it when required.
Moreover, the double-nozzle design we suggest in the JetRIS setup at GSI, whose main parameters are listed in Table 1, can be effectively used at different vacuum pump capacities (see Section 3.4 above).
To help the authors of works [32,33] prepare for the future online measurement of nobelium isotopes, we also performed the Monte Carlo simulations for the 253No atomic beam in the argon supersonic jet. The results of these calculations, which are presented in Section 3.7, allow us to quantitatively estimate the efficiency of the JetRIS setup operation using the proposed double-nozzle technique.
Implementing the proposed double-nozzle technique to the JetRIS setup will be enough to replace the present GSI nozzle with the double-nozzle system presented in the article.
In conclusion, we can say that the double-nozzle design, which main parameters are listed in Table 1, can also be used for other similar setups or/and projects. If, e.g., operation conditions or specific requirements to these setups/projects are very different from that in the JetRIS setup, then we can make similar computer experiments to find an optimal solution for these cases upon the corresponding request.