Experimental study of ECR plasmas in stationary versus turbulent regimes for in-plasma $\beta$-decay investigations

Abstract

A new experimental study of electron cyclotron resonance (ECR) plasmas in stationary versus turbulent regimes was recently carried out in the framework of the PANDORA (plasmas for astrophysics, nuclear decays observation and radiation for archaeometry) project, which aims to measure $\beta$-decays of nuclear astrophysical interest for the first time in laboratory plasmas emulating some stellar-like conditions. The experimental study was finalized to correlate the nuclear activity to the thermodynamic conditions of the plasma environment by means of a multi-diagnostic approach. Advanced analysis methods and non-invasive tools were properly developed, including high-resolution space-resolved X-ray spectroscopy and imaging, which allow unprecedented investigations of thermodynamic plasma properties as well as of plasma structure, confinement and dynamics of losses. Furthermore, an innovative plasma heating mode (two-close-frequency heating) has been investigated, showing to be a powerful method for improving the plasma stability, damping instabilities. In order to correlate plasma properties to nuclear activity, GEANT4 simulations devoted to design a $\gamma$-ray detector array to tag the in-plasma $\beta$-decays by detecting $\gamma$-rays emitted by daughters nuclei were carried out. The sensitivity of the PANDORA setup was specifically investigated in a virtual experiment, for the first PANDORA physics cases: $^{134}$Cs, $^{176}$Lu, $^{94}$Nb. The obtained results allowed to demonstrate that laboratory ECR plasmas in compact traps are suitable and interesting environments for fundamental studies as well as for applications.

1 Introduction

Plasmas in laboratories can be generated by the ECR heating method, through electromagnetic waves interacting with gases or vapours in presence of a well-shaped magnetic field. Magnetized ECR plasmas in compact traps, when used in quality of ion sources (ECRIS), are able to produce high intensity beams of highly charged ions, supporting the growing interest in the areas of both fundamental science (nuclear and particle physics especially) and applied research. On the other hand, they can become a very useful and interesting experimental environment for fundamental studies of nuclear astrophysical interest. In fact, theoretical predictions have shown that the ionization state of the in-plasma isotopes can modify, even of several orders of magnitudes, the isotopes lifetimes [1, 2], due to mechanisms such as the bound state $\beta$-decay [3, 4], where the electron emitted is captured in one of the inner shells of the atom. In neutral atoms, or ions with only a few holes in the atomic levels, such decay mechanism is hindered because the emitted electron is unable to access a bound final state, whilst it becomes possible in highly ionized ions. Studies related to fundamental physics of nuclear decays in an astrophysics-like environment are expected to have a major impact in the study of nuclear astrophysics processes (BBN, s-processing, CosmoChronometers) [5,6,7,8] and only few experimental evidences were collected, up to now, using the storage rings (SR). These measurements showed that the half-life of fully stripped $^{187}$Re$^{75+}$ ions collapsed of 9 orders of magnitude [4] with respect to the neutral $^{187}$Re atoms ($T_{1/2}\sim$ 42 Gyr), or that bare $^{163}$Dy$^{66+}$ nuclei, stable as neutral atoms, became radioactive with $T_{1/2}\sim$ 33 days [9]. A better comprehension of this phenomenon and further investigations are still necessary. In this work, magnetized ECR plasmas in compact traps are investigated as experimental environment for $\beta$-decays investigations, in the frame of the PANDORA project [10, 11], which proposes a radically different and complementary approach to those attempted in the past.

1.1 ECR plasma for $\beta$-decays nuclear astrophysics studies

The PANDORA project, supported by INFN, aims at measuring the variation of the half-life of $\beta$-decaying nuclei, due to a high degree of ionization of the atomic species investigated, in a magnetically confined ECR plasma able to emulate some stellar-like conditions. In the PANDORA trap, we plan to build [12], the radionuclides can be trapped in a dynamic-equilibrium, in a magneto hydro dynamically (MHD) plasma, living for weeks with on-average constant local electron density $n_e$ and electron temperature $T_e$ (the plasmas reach $n_e\sim 10^{11}$–$10^{13}$ cm$^{-3},\ T_e\sim 0.1$–$100\,$keV). In ECR trap, the ions are cold (ion temperature is always below 1 eV). The plasma is confined by multi-Tesla magnetic fields and resonantly heated by some kWs of microwave power in the 2.45–28 GHz frequency range. A high density of hot electrons can be achieved in ECR plasma by optimizing the magnetic confinement, resonance field gradients, microwave frequency and power, as well as the neutral flow [13]. Also the charge state distribution (CSD) can be modulated according to the pumping RF power, the magnetic field and the background pressure. First calculation at ECRIS densities and temperatures confirm that even in non-LTE (non-local thermodynamic equilibrium) conditions (due to the low density) the abundances of charge states are for many of the selected physics cases very similar to the ones occurring in astrophysical conditions [14, 15]. This makes the experiments running in an ECR plasma trap directly scalable to astrophysical systems. Figure 1 shows the expected lifetime versus $T_e$ at densities of PANDORA plasmas, predicted by Takahashi and Yokoi [2] (green line) and by the PANDORA collaboration for n-LTE plasma (blue line), for $^{176}$Lu (on the left) and for $^{134}$Cs (on the right).

Fig. 1

Expected lifetime vs. $T_e$ at densities of PANDORA plasmas, for $^{176}$Lu (left) and $^{134}$Cs (right) (courtesy of A. Mengoni). The black line shows the lifetime values of the neutral isotopes. Green and blue lines show the trend of the lifetime variation with $T_e$ predicted by Takahashi and Yokoi [2] and by the PANDORA collaboration for n-LTE plasma, respectively

The PANDORA experimental approach is based on maintaining plasma in a dynamical equilibrium for long periods of time, extending to months. The isotope can be directly fluxed inside the chamber to be turned into plasma-state with relative abundances of isotope densities versus a buffer plasma (created by He, O or Ar up to $n_e\sim 10^{13}$ cm$^{-3}$) reaching 1:100. Whilst the isotopes decay, the daughter nuclei confined in the plasma emit $\gamma$-rays (at energy of hundreds of keV) that can be detected by a proper $\gamma$-detector array surrounding the magnetic trap. The total amount of decays N can be determined as follows:

$$\begin{aligned} N(t_{\text {m}})=\lambda n_iV_pt_{m} \end{aligned}$$

(1)

where $\lambda$, $n_i$, $V_p$, $t_{m}$ are, respectively, the isotope’s activity, the ions density, the plasma volume and the total measurement time. Due to dynamical equilibrium, $\lambda$ $n_iV_p$ is constant in time and N linearly increases versus $t_{m}$. Any physical condition (combination of $n_e$ and $T_e$) in non-LTE determines a certain CSD of the in-plasma ions, similarly to what happens in the stars interiors (where, however, the plasma in under LTE equilibrium).

In order to deconvolve $\lambda$, plasma parameters must be known and online monitored and since $n\equiv n(x,y,z)$, its spatial dependence must be also determined.

In summary, for the future PANDORA measurements it is mandatory to have:

• Stable and reproducible plasma conditions;

• An accurate online monitoring of the plasma properties and parameters;

• A method to tag the $\beta -decaying$ isotopes of interest.

Several activities, carried out at INFN-LNS and in collaboration with the ATOMKI Laboratories, aimed to characterize thermodynamic properties of ECR plasmas, mitigating kinetic turbulences naturally occurring in these plasmas. The experimental study here presented consists in three main synergic sub-activities (see Fig. 2): (i) an experimental study to investigate a new plasma heating method able to improve the plasma stability; (ii) an experimental study devoted to characterize thermodynamically the plasma environment through new diagnostic tools and techniques; (iii) GEANT4 simulations to design the detection system for measuring $\beta$-decays via $\gamma$-rays tagging.

Fig. 2

Summary of the three main synergic sub-activities of the experimental study

This paper is organized as follows: Sect. 2 describes the multi-diagnostic setup, with details about the X-ray spectrally resolved imaging and the frequency-resolved spectroscopy device. Section 3 shows the analytical methods developed to analyse the X-ray pin-hole images (in single-photon-counted SPhC) and the signals collected by the two-pin RF probe. The whole set of the experimental results is described in Sects. 4 and 5, highlighting the original impact on the plasma physics research. Section 6 reports on the GEANT4 simulations implemented for supporting the $\gamma$-ray detector array design.

2 Experimental setup

The direct correlation of the plasma environment and the $\beta$-decay activity can be done by simultaneously identifying the photons emitted by the plasma (from microwave to hard X-ray) and $\gamma$-rays emitted after the isotope’s $\beta$-decay. Setup and tools were developed for performing multidisciplinary studies. They will be installed around the ECR plasma trap and used simultaneously.

2.1 PANDORA test bench: the ECRIS at ATOMKI

Since the design of the PANDORA plasma trap is ongoing, measurements were collected on a downsized test bench, i.e. the ECRIS at the ATOMKI Laboratory [16] (Fig. 3a).

The ATOMKI-ECRIS is a classical, room-temperature second-generation ion source. Its basic operation frequency is 14.25 GHz supplied by a klystron amplifier. A second frequency can be coupled in by the same WR62 waveguide supplied by a TWT amplifier, operating in the range 13.6–14.6 GHz [17]. The ion source is not connected to accelerator; thus, it is highly applicable for plasma physics investigations. Further technical and application details are in a summary paper [18]. The magnetic field necessary for plasma confinement is produced by employing a magnetic system (as typical for ECR ion sources) consisting of a sextupolar permanent magnet (for radial confinement) nested inside two solenoidal coils (for axial confinement) (Fig. 3b). This magnetic system configuration is called minimum-B and allows the confinement of a plasma located around the plasma chamber axis, providing magnetohydrodynamical (MHD) equilibrium and stability. For the ATOMKI-ECRIS, the axial magnetic field is 1.26 T (injection), 0.39 T (minimum) and 0.95 T (extraction). The hexapole produced magnetic injection at the plasma chamber wall (R=29 mm) is 1.2 T. During all the measurements, a middle-charged argon plasma was kept in the ion source with fixed gas dosing valve and with fixed (maximal) magnetic field. Representatives of the charge state distribution of the extracted ion beam (Ar$^{6+}$, Ar$^{9+}$, Ar$^{11+}$) were continuously monitored which gave information on the ionization efficiency and on CSD shift.

Fig. 3

a The ATOMKI-ECR ion source. b The axial magnetic field distribution of the Atomki-ECRIS in the $B_{\text {min}}$ magnetic field configuration

An new design of the plasma chamber (210 mm of length and 58 mm of diameter) was used. The chamber wall was covered with a Ta liner, the injection and extraction plate were made of Al and Ti, respectively. About 2/3rd part of the injection plate was made of Al mesh ($400\,\mu$m of wire diameter) providing closed resonant cavity and around 60$\%$ transparency for X-ray imaging. The ion source, not connected to any accelerator, is suitable for plasma physics investigations. It was optimized for middle-charged argon ion production. The gas pressure measured at the injection box of the ion source was $2 \cdot 10^{6}$ mbar and ion beam was extracted from the plasma by 10 kV extraction voltage.

2.2 The multi-diagnostics setup

In ECR plasmas confined in $B_{\text {min}}$ structures, the energy of the electrons ranges from a $few\ eV$ to hundreds of keV or even 1 MeV and, since ECRIS plasmas are in non-LTE condition, the typical electron energy distribution function (EEDF) consists of three different populations: the hot ($T_e \ge 100$ keV), the warm ($T_e$ from 100 eV up to tens of keV) and the cold ($T_e \sim$ 1–100 eV) one. For a complete characterization, a multiplicity of tools and analysis methods become necessary and a multi-diagnostic setup [17, 19,20,21] (sketched in Fig. 4) was properly developed to be installed, in perspective, in PANDORA. A subsystem of this setup was used to obtain the experimental results described in this work. In particular, the two techniques simultaneously employed are the X-ray pin-hole CCD camera and the two-pin RF probe tools (highlighted in panels in Fig. 4) and will be described in detail in the next subsections.

Fig. 4

Sketch of the multi-diagnostic setup consisting in a collection of non-invasive tools

2.2.1 The two-pin RF probe and spectrum analyzer

The two-pin RF probe [22, 23], designed to detect the EM plasma emission in GHz domain, was installed inside the plasma chamber through one of the gas input holes in the injection plate (Fig. 5, left). It was flexible with an outer diameter of 4 mm, a pin length of 3.5 mm and a pin distance of 2 mm.

Fig. 5

Left—Blocks diagram of the system with the two-pin RF probe connected to the spectrum analyzer. Right—Frequency-resolved spectrum of the detected power inside the plasma chamber

The spectrum analyzer (SA) operated with a frequency span in the range 13–15 GHz with a resolution bandwidth (RBW) of 3 MHz, and a sweep time of 400 ms. The probe acquisition setup consists also of a RF power limiter, 10 dBm limiting and operating in the range 10–26.5 GHz, and a set of variable attenuators in the dynamical range 3–30 dB. Connecting the two-pin RF probe to the SA, it was possible to characterize the EM emission in GHz ranges [20]. A blocks diagram of the system (left) and a typical spectrum (right) are shown in Fig. 5.

This setup is able to detect both the main pumping frequency (@ 14.05 GHz) and self-plasma emitted radiations, consisting in several sub-harmonics. Since the RF plasma self-emission sub-harmonics can represent signatures of plasma kinetic instabilities (characterized by fast RF and X-ray bursts [24,25,26,27]), this tool can be used to detect and characterize turbulent plasma regimes, in order to: (a) find a way to damp them and, consequently, improve the ECR plasma stability and ECRIS performances [28, 29]; (b) reproduce and study these interesting phenomena of astrophysics interest.

2.2.2 The X-ray pin-hole camera system

The soft X-ray diagnostics allow to probe the warm plasma population, which plays a crucial role in ionization and charge distribution build-up. Several diagnostics were developed over the years to perform volumetric X-ray spectroscopy [30,31,32,33,34,35,36]. High-precision X-ray measurements can be also carried out in a spatially resolved way by means of the powerful X-ray pin-hole camera tool. Even if our teams have already been using the X-ray camera techniques combined with pinholes for more than 15 years [37,38,39,40], and this method has been routinely used in several other laboratories (for example, at Z-machines [41, 42] or in plasma fusion devices [43,44,45]), recently, both the tool [46] and the advanced analytical method [47] have been drastically improved.

The diagnostic setup used to perform high-resolution X-ray imaging [46] is shown in Fig. 6A. The CCD X-ray camera (model Andor, iKon-M SO series) was made of 1024$\times$1024 pixels, with an optimal quantum efficiency in the range 0.5–20 keV. It was coupled to a Pb pin-hole (thickness 2 mm, diameter $\Phi$=400$\mu$m) placed along the axis, facing the chamber from the injection flange. A Ti window (9.5 $\mu$m of thickness) was used to screen the CCD from the visible and UV light coming out from the plasma.

Fig. 6

A Sketch of the X-ray pin-hole CCD camera system. On the right, the plasma chamber highlighting the different materials and relative X-ray fluorescence peaks. The sketch also shows (in white and yellow) the magnetic field lines going from the plasmoid towards the chamber walls: electrons are shown as bullets, red for the ones impinging on the lateral walls and yellow for the ones flowing towards the endplates. B A X-ray image (in logarithmic scale) shows magnetic branches and poles (radiation coming from the extraction plate and from the lateral chamber walls), the extraction hole and the plasma emission. C Perspective front view of the plasma chamber from the same side as the X-ray pin-hole camera

To study plasma kinetic instabilities (typically triggered above certain power thresholds [25, 48]), we needed to work with the ECR plasma heated at a high power. Thus, the overall pin-hole system was redesigned, including a multi-disc Pb collimator for an extra shielding of scattered radiation affecting the image quality. The multi-disc collimator consists of two lead discs with the same thickness (1 mm) and different diameters $\Phi$ (2 mm, 1 mm). Other details are described in Ref. [46]. The magnification was $M=0.244$. In this system, the pin-hole camera was operated up to 200 W of total incident power in the plasma—one order of magnitude higher than previous measurements (where the pumping power was limited to 30 W [49, 50]). In this way, both stable and turbulent plasma regimes were investigated, as well as new plasma heating methods [28, 29, 51].

The sketch of the ATOMKI plasma chamber is illustrated in Fig. 6A (on the right). It reports the ellipsoidal shape of the plasma core, the so-called “plasmoid”, which is typically contained within the iso-magnetic surface fixed by the ECR condition $\omega _{\text {RF}}=qB/m$, where $\omega _{\text {RF}}$ is the RF pumping frequency, B is the magnetic field, and q and m are the electron charge and mass.

From this high-density core, fluxes of deconfined electrons and ions escape, according to the so-called “loss cones” [52]. The field lines lying inside the loss cones are represented in white and yellow. Small bullets of different colours represent the lost electrons moving towards the lateral chamber walls (red dots) or towards the endplate (yellow dots). The perspective front-view of the plasma chamber (i.e. the same view as the pin-hole camera setup) is sketched in Fig. 6C. The Al mesh is clearly visible. A simultaneous spatial and energy resolution of, respectively, 0.4 mm and 0.326 keV at 8.1 keV was reached.

3 Analytical methods: X-rays and RF analysis

In this section, two analytical methods properly developed from the ground using a suite of MATLAB codes are described. The first algorithm (Sect. 3.1) was finalized to analyse the raw data acquired using the X-ray pin-hole camera tool in order to obtain high-resolution spatially resolved X-ray spectroscopy through single-photon-counting (SPhC) analysis. It represents a powerful method for plasma structure evaluation and for locally determining plasma density and temperature. The second algorithm (Sect. 3.2) was finalized to study the plasma kinetic turbulences, for the first time in a quantitative way; the method provided a new parameter that defines the strength of instabilities.

3.1 X-ray spectrally resolved imaging algorithm

When a photon impinges in the CCD, a charge proportional to the deposed energy is generated; the analogue digital unit (ADU) loaded in each pixel is proportional with the product of the number of incident photons and their energies. There are two main operative modes: the spectrally integrated (SpI) and the SPhC one. In SpI mode, the images are obtained with long exposure time $t_{\text {exp}}$ (tens of seconds, as the one shown in Fig. 6B), thus the CCD camera registers multi-events, losing the spectral information of the incoming X-rays. Whilst, the SPhC mode allows to decouple the number of photons from their own energies, performing local energy determination and more powerful investigations than SpI one. To obtain the SPhC mode, a short $t_{\text {exp}}$ (tens of ms) must be fixed for each image and thousands of frames must be recorded. The SPhC minimizes the probability that two (or more) photons hit the same pixel during the $t_{\text {exp}}$ of a single frame with a consequent energy information loss. A processing procedure, made of the follow 4 steps, was developed [47]:

(I) Grouping process (Gr-p): even setting small $t_{\text {exp}}$, one or more photons can release their energy in a cluster of pixels. To assure the proportionality ADU versus single-photon (sPh) energy, the charge of the given cluster must be assigned to a sPh-detection event. Overlapped clusters are spurious events and must be discarded. Thus, a parameter (called S), which represents the maximum cluster size (in pixels) that can be considered as a sPh event, was introduced. The code scans pixel-by-pixel each image and processes each group of neighbouring pixels. Clusters having size $>S$ are filtered out, the other ones are grouped reconstructing the total charge due to a sPh event. At the end of the scan of all the image frames, a set of data (X, Y, E, $N_{t}$) is obtained, where $N_{t}$ is the total number of photons with energy E in the position (X, Y). The plot of $N_{t}$ versus E provides the X-ray spectrum in the full-frame CCD. The impact of Gr-p is shown in Fig. 7a, which compares the raw spectrum (green) with the processed one (black). In the raw spectrum, no fluorescence peak can be identified (overlapped clusters worsen the overall energy resolution), whilst the Gr-p enables characteristic fluorescence peaks identification.

Fig. 7

a X-ray spectrum evaluated before (green) and after (black) having implemented the Gr-p. b Calibrated X-ray spectra for five different values of the input S parameter

(II) Energy filtering process: Fig. 8a shows a SPhC image obtained by processing 4200 image frames at $t_{\text {exp}}=0.5$ s. The energy-filtered imaging consists in selecting only those pixels whose energy corresponds to a given $\Delta$E, i.e. getting image about the distribution of that single element fluorescence.

Fig. 8

a Energy-unfiltered X-ray image. Energy-filtered image: Ar (b), Ti (c) and Ta (d)

The application of an energy filter in the range below $K_{\alpha ,\beta }$ lines of Ar provides imaging of Ar plasma only (Fig. 8b). Similarly, filters applied to Ti or Ta lines allow to visualize the X-rays coming from the chamber walls (Fig. 8c, d), i.e. the bremsstrahlung and fluorescence X-rays coming from, respectively, electrons impinging axially and radially on plasma chamber walls.

(III) High dynamical range (HDR) analysis: the images shown in Fig. 8 display a strongly inhomogeneous intensity distribution. Fixing a given $t_{\text {exp}}$, some areas can be over- or under-exposed, since the emission rate is space-dependent. In addition, the Gr-p acts differently in different regions: ROIs where the flux is particularly intense might result not SPhC and most of the clusters would be discarded by the algorithm. An example is given by rectangles in Fig. 9a, d (respectively, for the Ar- and Ti-filtered image): the magnetic branches (green squared) and poles (yellow squared), due to the radiation coming from deconfined electrons in the chamber walls, are dark at $t_{\text {exp}}=0.5 s$ since not SPhC. HDR analysis fixes this problem; it consists in acquiring the image at different $t_{\text {exp}}$, producing a weighted convoluted image.

Fig. 9

Ar-filtered image post-processed by: Gr-p (a), HDR (b), HDR RON-r (c) procedure

We collected 4200 frames at $t_{\text {exp}}=0.5$ s and 1000 frames at $t_{\text {exp}}=0.05$ s, and the HDR result is displayed in Fig. 9b, e, containing information either in the plasma regions (white rectangles), branches and poles. Since the degree of the missing information is spatial-dependent, the implemented algorithm operates by exploring any pixel in each image frame, defining a proper “mask”. In such a way, only pixel clusters that in the acquired image were not photon-counted (thus filtered out by the Gr-process) were replaced by the ones taken by HDR procedure, once weighted by the HDR mask. Details are reported in Ref. [47].

(IV) Read out noise (RON) removal (RON-r): readout time $t_{\text {RO}}$ was around 1 ms so that in case of $t_{\text {RO}}\simeq t_{\text {exp}}$ the CCD also collected a not negligible number of photons whilst the charges were transferring to the readout register. This produced an artefact in the images, a kind of strips as shown in Figs. 8 and 9a, b, d, e. Our algorithm was also optimized to remove this RON. RON-r post-processed HDR image is shown in Fig. 9c, f, showing a dramatic improvement of the signal-over-noise ratio and spatial resolution.

3.2 RF spectral analysis

The kinetic plasma instabilities are caused by a strongly anisotropic distribution of the plasma electrons in the velocity phase space; the strength of the instability depends on the velocity space for which $\frac{\delta f(v_{\perp })}{\delta v_{\perp }} > 0$. Instabilities cause the emission of microwaves bursts and the energy of fast electrons may be released as coherent (maser) electromagnetic radiation due to development of electron cyclotron instabilities. The emission frequency depends on the ECR frequency, i.e. on local magnetic field B.

Fig. 10

SA collected spectra for: A stable plasma regime whose power spectrum is characterized by only RF pumping peak without self-emitted radiation; B unstable plasma regime whose power spectrum is characterized by many sub-harmonics at low power; C unstable plasma regime whose power spectrum is characterized by sub-harmonics both at high and low power

Since plasma kinetic instabilities are characterized by fast RF and X-ray bursts, the plasma self-radio emission in the GHz range can be considered as a signature of the plasma instability regime [24, 25]. Thus, plasma instabilities are detectable and quantifiable observing their effect on the corresponding RF power spectra.

The method allowed to semiempirically define a new parameter, called $I_S$ (Instability Strength), able to give for the first time a quantitative estimation of the instabilities strength. Three typical experimental spectra, acquired using a pumping power of 200 W and different single pumping frequency, are reported in Fig. 10. The case (A) shows the power RF spectrum of a typical stable plasma regime, characterized only by the pumping RF peak (@ 14.25 GHz) and there is not plasma self-radio emission. The cases (B) and (C) show power spectra typical of plasma unstable configurations: in addition to the main peak, plasma self-emissions appear. The instability strength is related both to the amplitude of the sub-harmonics and to their frequency spread; thus, both the integral and the broadening of frequency spectra must be considered. The $I_S$ was defined as follows [29]:

$$I_{S} = \left( {\int\limits_{{13{\kern 1pt} {\text{GHz}}}}^{{15{\kern 1pt} {\text{GHz}}}} {\frac{{{\text{d}}P(f)}}{{{\text{d}}f}}{\text{d}}f - P_{{{\text{mp}}}} } } \right)\left( {1 + w\left( {N_{{{\text{sub}}}} - 1} \right)} \right)$$

(2)

where $P_{\text {mp}}$ is the integral of the power of the main peak of pumping frequency, $N_{\text {sub}}$ the number of sub-harmonics and w a weight factor. $I_S$ was calculated considering the amplitude (integral of the power) of RF plasma self-emitted signal, once subtracted the main pumping wave contribution; this number has been then multiplied by a factor which considers the number of sub-harmonics $N_{\text {sub}}$ with a proper weight factor w, which was optimized and set at 0.1. Experimental evidences demonstrated that $I_S$ parameter follows in a reasonable way what happens in the plasma in unstable conditions, i.e. it is expected that plasma instabilities increase with the power [25, 48] and a clear increase of $I_S$ versus power was measured. An $I_S$ value was assigned for each configuration, to correlate it with the operative parameters and plasma heating mechanisms.

4 Experimental results: plasma stability and confinement improvement by TCFH

The instability threshold depends principally on the strength of the magnetic field in terms of $\frac{B_{\text {min}}}{B_{\text {ECR}}}$ [53, 54] and from other parameters as RF pumping power and pressure [55]. The exact mechanism of turbulent regimes of plasmas is still unknown and a deeper investigation is necessary. The study described in this section aims to investigate the role of the TCFH on plasma stability and confinement, when the two frequencies gap is consistent with the estimation given by Ref. [56], under which the two resonance zones are overlapped. The main theoretical aspects of the quasi-linear theory of two frequency heating used to give a robust explanation of TCFH consequences on the heating and confinement dynamics were discussed in Ref. [56]. Summarizing, in TCFH, an higher diffusion in the parallel direction of the particles’ energy $v_\parallel$ should limit the energy that the plasma can self-couple to the electromagnetic modes supporting the instability. This is the basic assumption that will be validated by experimental results in the following sections. Moreover, recent studies [25] predicted that turbulent regimes of plasma can generate precipitations of the energetic electrons in the loss cones (producing the X-ray burst emission when they impact in the plasma chamber walls) thus increasing the losses compared to the plasma emission. Our study can play a relevant role in order to verify this hypothesis, thus trying to experimentally correlate the release of energy via RF and, especially, X-ray bursts with the onset of the instability.

4.1 TCFH impact on plasma stability

Plasma instabilities were detected via RF spectra and characterized by $I_S$ parameter. At first the system was characterized in single-frequency heating (SFH) mode, then in TCFH mode. To validate the $I_S$ consistency, plots concerning the $I_S$ behaviour versus the RF power were made. The definition of $I_S$ resulted to be consistent, increasing steadily with the power, as expected.

Fig. 11

a Comparison of the $I_S$ parameter for SFH and TCFH, at fixed power. TCFH damps the instability almost over the entire frequency set. b Spectral structures as detected by the SA at different frequencies during the SFH scan; the image was plotted in logarithm colour scale. c Histograms showing the RF power provided in SFH (only TWT @ 13.90 GHz) or in TCFH mode by the two generators (TWT @ 13.90 GHz + Klystron @ 14.25 GHz). Green dots are the relative $I_S$ factor. The combination of two frequencies is more stable than a single one, even if the power was increased of almost a factor 2

Moreover, the frequency tuning, both in SFH and in TCFH mode, was systematically explored. The results are shown in Fig. 11a. Considering SFH, $I_S$ mirrors the strongest instability at the lowest frequency we used, i.e. 13.60 GHz. Even in this case, this is in agreement with the general conditions of the source (at 13.60 GHz it has been obtained the largest high-energy X-ray flux and the largest total X-ray dose [28]). This is also consistent with the fact that at 13.60 GHz the source is operated well above the $\frac{B_{\text {min}}}{B_{\text {ECR}}}$ threshold that is universally considered as the trigger for the instability onset [53, 55].

Figure 11b shows the RF probe detected signal analysed through the SA, where it is possible to observe the characteristic sub-harmonic peaks. The pumping frequency lies along the line y=x, thus the “drop” below it gives a qualitative indication about the instability strength which is not only related to the amplitude but also to the frequency spread of the self-generated sub-harmonics. It is possible to note that instabilities generate sub-harmonics for all the pumping frequencies and the emitted radiation is predominantly at lower frequencies than the plasma heating frequency (as measured also in [24]). A 3D plot of the RF probe detected signal, equivalent to the pseudo-colour plot shown in Fig. 11b, is shown in the panel on the top in Fig. 4.

The most relevant result is shown in Fig. 11a: it is possible to observe that $I_S$ drops dramatically operating in TCFH, confirming that the TCFH is able to damp the instabilities almost over the entire investigated frequency spectrum. Figure 11c highlights the instability damping at 13.90GHz and 200 W, which was a very unstable regime. It is clear that at this operative frequency plasma is highly unstable already at low power levels. Anyway, the addition of the second wave damps the instability even if the total amount of power reaches 200 W (120 W by TWT + 80 W by Klystron). This means that the instability can be damped by TCFH even if the second frequency brings additional RF power into the system. It is a very important result both for multidisciplinary and fundamental physics (in PANDORA plasmas must be maintained stable for weeks) and accelerator physics (the plasma turbulence damping can improve the ion sources performances [53, 55]).

4.2 TCFH impact on plasma confinement and losses

Investigations on plasma confinement and dynamics of electron losses, studying the effect of the TCFH, were performed through the X-ray imaging [51]. These studies were carried out operating in spectrally integrated mode (collecting images with 50 s of $t_{\text {exp}}$) because too long acquisition times are needed to carry out a complete characterization in SPhC mode: to obtain one single SPhC image (in our set condition) around 4–5 h are required.

We introduced the $\frac{L}{P}$ ratio, defined as the sum of counts in the magnetic branch plus pole regions (the losses L) divided by the sum of counts in the plasma ROIs (the plasma contribution P). This ratio has been mathematically labelled as $R_{\text {LP}}$. The number of counts detected by the X-ray pin-hole camera is normalized for each ROI area (the selected ROIs are shown in Fig. 6b: in white, blue and green, respectively, for plasma, magnetic branches and poles).

Figure 12a shows the trend of $R_{\text {LP}}$ and $I_S$ versus the RF power for SFH. Both parameters increase as a function of the power, thus a direct correlation can be argued. A similar trend is observed also in the case of TCFH.

Fig. 12

Trend of the $R_{\text {LP}}$ (red) of total counts detected by the X-ray pin-hole camera vs. SFH power scan (a) and SFH frequency scan (b). $I_S$ parameter is also plotted in blue

In order to quantitatively evaluate the degree of correlation, $R_{\text {LP}}$ versus $I_S$ plots were performed. The correlation is confirmed by the R Pearson parameter, being $R=0.93$ and $R=0.75$, respectively, for SFH and TCFH configurations. $I_S$ and $R_{\text {LP}}$ as a function of different pumping frequency in SFH were also measured, even if the correlation plots show no correlation in both cases. Only performing two different correlation plots of the data distinguishing frequencies lower than 13.90 GHz and higher than 13.90 GHz, we find a weakly correlation in SFH for lower frequencies ($R=0.65$) but not for higher frequencies ($R=0.09$). This can be due to the fact that above the critical threshold, the turbulent coefficient dominates on the collisional one, thus the losses are regulated by the turbulence (electronic precipitation from the magnetic trap).

In order to verify if the plasma density distribution is rearranged by TCFH, becoming denser in the central region of the plasma chamber (where the B-field is lower), the counts measured in the plasma ROIs at different radial positions were analysed. A parameter (labelled as K) to describe the degree of the centralization of the plasma was introduced. It is calculated as the ratio of the counts of the central (black ROI in Fig. 6b) and the side (white ROIs in Fig. 6b) regions of the plasmoid. The higher is K, the plasma is more centralized. K was estimated both for single and double frequency scans. As an example, the K and $I_S$ parameters are plotted in Fig. 13a, for SFH and TCFH at 13.90 GHz, 14.25 GHz and 13.90 GHz + 14.25 GHz (200 W).

Fig. 13

a K and $I_S$ parameters for SFH and TCFH modes at 13.9 GHz, 14.25 GHz and 13.9 GHz + 14.25 GHz (@ 200 W). b Comparison of K parameter in SFH vs. TCFH mode

As a consequence of the second frequency, it becomes about $10\%$ more centralized. Moreover, K parameter was calculated for SFH and TCFH scans, comparing them at those frequencies where the instability strength was pronounced (i.e. $I_S$ parameter is higher than 7 mW) at SFH mode. These frequencies are 13.60, 13.80, 13.85, 13.90, 14.00, 14.05 GHz. The corresponding centralization parameters are plotted in Fig. 13b. This figure, together with Fig. 11a, clearly demonstrates that in TCFH mode the instability is effectively damped by the second frequency; meanwhile, the plasma is rearranged to be denser in a central region of the plasma.

5 Experimental results: plasma structure and thermodynamic characterizations by SPhC

Deeper investigations can be performed by SPhC analysis, allowing X-ray imaging of the elemental distribution (distinguishing the emission coming from each material), space-resolved spectroscopy (determining the elemental composition in each ROI) and spectrally resolved imaging [57, 58]. It is also possible to study the plasma structure and measure the plasma radius. Two complementary analyses can be simultaneously performed: the HDR Energy-filtered Imaging (Sect. 5.1) and the HDR Space-filtered Spectroscopy (Sect. 5.2). The experimental configuration in SFH at a pumping frequency of 13.90 GHz and a total net power of 200 W was investigated in SPhC mode. The analysis of other configurations in SPhC, both for TCFH and SFH, is ongoing and results will be investigated and compared to this one in the next future.

5.1 HDR energy-filtered Imaging

The HDR energy-filtered imaging consists in selecting only the pixels of the frames which are loaded by the photons having energy corresponding to a well-defined energy range. It allows the imaging of the elemental distribution. We present the HDR results obtained by selecting the energy ranges of Ar (2.96, 3.19 keV), Ti (4.51, 4.93 keV) and Ta (8.15 keV) fluorescence peaks. The images are shown in Fig. 14a and b, respectively, in red, blue and green pseudo-colour scale. A total of ten ROIs were selected in each image (shown in Fig. 15a) and the corresponding total counts of each ROI are shown in Fig. 14c, d, e, respectively, for Ar-, Ti-, Ta-filtered image.

Fig. 14

a Sketch of the plasma chamber highlighting the energy-filtered images of photons coming from Ar plasma (red) and from the metallic layers covering plasma chamber: Ti (blue) and Ta (green). b Merging of the three energy-filtered SPhC images. c, d, e Total counts of each ROI, respectively, for the Ar-, Ti- and Ta-energy-filtered images

For a quantitative analysis, we defined as $\xi _{k}=\int _{A_k}I_{i,j}$ the number of counts in a given k ROI, by integrating the intensity I of each i, j pixel in the ROI area $A_k$. Analysing the Ar-filtered Fig. 14a, it is possible to observe that the plasma emission in the ROI$_1$, i.e. in the plasma core, is higher than the one in the ROI$_2$ and ROI$_3$, i.e. in the plasma side. Since the ratio between ROI$_1$ and the mean of ROI$_2$ and ROI$_3$ emissivities is $\xi _{1}/(<\xi _{2},\xi _{3}>)\simeq 1.13$ there is a slight concentration of the plasma in the axial region. The high $\xi _{4}$ in Fig. 14a points out a strong plasma deconfinement along the magnetic branches. For a more accurate analysis of the spatial structure of the plasma, we draw a 1D intensity profile: it was taken along a “row ROI” of [$100\times 1024$] pixels (see Fig. 15b). This was useful to evaluate the plasma radius, that is $R_{\text {pl}}=15.53\ \pm \ 0.87$ mm, in agreement with the expected radius of the plasmoid (15.50 mm) just assuming the magnetic field profile.

Fig. 15

a HDR Ar energy-filtered image, including the ten selected ROIs. b The intensity profile evaluated along some “row ROI” of [$100\times 1024$] pixels (shown on the top)

Finally, the $R_{\text {LP}}$ parameter was also evaluated by analysing the SPhC image. An energy-filtered image with $E>3.80$ keV was considered to counts only the Ti and Ta fluorescence lines and Bremsstrahlung contributions, i.e. the total losses. The plasma emission was evaluated summing the counts in plasma ROIs of the Ar energy-filtered image, whilst the total losses emission summing the counts in the magnetic branches and poles ROIs of the image energy-filtered with $E>3.80$ keV. The measured $R_{\text {LP}}$ is 15.7. It is slightly higher than the one estimated by the SpI (for the same configuration) but still compatible.

5.2 HDR space-filtered spectroscopy

A complementary analysis of the SPhC energy-filtered imaging is the Space-filtered Spectroscopy, which consists in selecting a “collection” of pixels of a ROI and evaluating the corresponding spectrum. The spectra evaluated in six of the above-defined ROIs (2, 4, 6, 7, 8, 9) are shown in Fig. 16a: ROI$_2$ is the plasma only emitted spectrum, ROI$_4$ is for the plasma that is flowing in the magnetic branch, ROI$_6$ is the spectrum emitted from the magnetic branches and ROI$_7$, ROI$_8$ and ROI$_9$ the ones emitted from the magnetic poles. It is possible to observe that the emission of characteristic lines is strongly ROI-dependent, since Ar-plasma emission occurs mainly in ROI$_2$ and ROI$_4$, whilst ROI$_4$ reports also very intense $Ti_{K_{\alpha }, K_{\beta }}$ lines as well as ROI$_6$, whilst the ROI$_7$ and ROI$_8$ are dominated by $Ta_{L_{\alpha }}$, $Ta_{L_{\beta }}$ and $Ta_{L_{\gamma }}$ fluorescence.

Fig. 16

a Spectra evaluated in six ROIs. b The histogram shows the total counts per second of each fluorescence lines (Ar$_{K_{\alpha }, K_{\beta }}$, Ti$_{K_{\alpha }, K_{\beta }}$, Ta$_{L_{\alpha }, L_{\gamma }}$) for each selected ROI

A quantitative estimation of the total counts per second (cps) emitted in each energy interval of interest was performed by fitting the main peaks of Ar, Ti and Ta. The results are shown in Fig. 16b. For Ar and Ti we report the sum of $K_{\alpha }+K_{\beta }$ emissions, for Ta the $L_{\alpha }+L_{\gamma }$ emissions were considered. This plot is particularly helpful to investigate the plasma concentration from the core (ROI$_{1-4}$) till the branches (ROI$_6$) and chamber walls (ROI$_{7-8-9}$). The plasma emission characterizes both the internal ECR surface, as expected by the magnetic confinement, both the external one (ROIs 5 and 6). This is the evidence of the deconfined plasma by the magnetic branches and it appears also in the more external region ROI$_6$.

Finally, it is important to mention that both the whole spectrum and the ROIs spectra can be analysed in order to locally estimate the plasma parameters, in terms of electron temperature and electron and ion densities. The plasma emitted spectrum can be reelaborated in order to measure the so-called spectral emissivity curve providing the total equivalent number of photons emitted in the given energy interval over the entire sold angle measured by the detector. For a precise detector’s solid angle estimation, dedicated Monte Carlo simulations are needed. After evaluating the emissivity for each measured spectrum, it is fitted by a theoretical curve, following models for X-ray spectra fitting like the approach described in Ref. [59]. Another analysis can be done focusing the attention to the Ar fluorescence. Introducing a proper estimation of the $K_{\alpha }$ ionization and radiation yield cross section [60], and of the density and energy spatial distributions by simulations [61, 62], it is possible to retrieve the local argon density. The results here presented are crucial since it is now possible to have reliable data for both the contributions (Bremsstrahlung and fluorescence), with high enough statistics, minimization of noise and spurious signals; the analysis for getting the plasma density and temperatures is ongoing.

6 Simulations for the $\gamma$-ray array design

Results described in the previous sections form “the first part” of the needed experimental setup for PANDORA. They will be devoted to perform the online monitoring of plasma parameters, to characterize the plasma environment and to assure plasma stability. In order to perform studies of $\beta -$decay, it is necessary to implement the second part of the setup, consisting of a $\gamma$-ray HPGe detector array for the decay products tagging, as shown in the following section.

It should be highlighted that one of the main targets necessary for PANDORA measurement is to obtain highly ion charged states. It is known that, for ECRISs, high charge state ions are primarily produced by sequential impact ionization, which means that the ions must remain in the plasma long enough (up to hundreds of ms) to reach high charge states. Therefore, one of the main parameters determining the performance of an ECRIS is the product of the plasma density, $n_e$ and ion confinement time $\tau _i$, called quality factor Q = $n_e$ $\tau _i$. In general, source development has followed the semiempirical scaling laws first proposed by Geller [52], which state that the plasma density scales with the square of the frequency $n_e \sim \omega _2^{\text {RF}}$. As the frequency increases, the magnetic fields must be scaled accordingly to fulfil the resonant heating condition for the plasma electrons [53]. As a consequence, the ion confinement time increases: in fact, it is proportional to the value of the so-called mirror ratio, i.e. the ratio between the maximum and the minimum magnetic field value for the considered magnetic system. Furthermore, the ion confinement time increases with plasma chamber length (mirror length) and radius, so these two values need to be increased if the objective is to obtain the highest possible charge states [63]. The performances of a magnetic system in terms of plasma confining capability are expressed by five magnetic field values: the value corresponding to the ECR, $B_{\text {ECR}}$, that depends on the heating frequency $f_{\text {RF}}$; the maxima values at the injection and extraction sides of the trap, $B_{\text {inj}}$ and $B_{\text {ext}}$, respectively, along the plasma chamber axis; the minimum, $B_{\text {min}}$, usually located at a central position along the plasma chamber axis; the maximum generated by the sextupole at the plasma chamber inner walls, $B_{\text {hex}}$. In superconducting traps, special attention must be paid to the minimum field, $B_{\text {min}}$: it has been experimental observed that, in order to obtain the highest electron density and to reach the optimal charge state, one has to have 0.65 < $B_{\text {min}}/B_{\text {ECR}}$ < 0.75 [30, 32]. If this ratio exceeds the upper value, sudden nonlinear effects arise, increasing the plasma X-ray emission and thus the heat load on the cryostat. All details about PANDORA trap design are described in [12].

6.1 Design of the PANDORA setup in GEANT4

GEANT4 simulations [64], focused on the design of the array of $\gamma$-ray detectors and on investigating the total efficiency $\epsilon _t$ of the array in terms of detectors type and of their optimal displacement around the trap (including collimation systems), were carried out [65, 66]. According to the trap design described in [12], the PANDORA system was implemented in GEANT4 (a sketch is shown in Fig. 17a). It consists of a stainless steel chamber (external radius of 150 mm, length of 700 mm and thickness of 10 mm, shown in grey) surrounded by a magnetic system made of three NbTi superconducting coils (in yellow) and in a NbTi six-conductor hexapole (in red). A single cryostat, containing the three coils plus the hexapole magnets, has been simulated as an Al cylindrical structure (in pink). Twelve azimuthal holes along the conductor hexapole interspaces were performed in the cryostat structure as well as in the plasma chamber. The idea is to use the hollowed cryostat as multi-collimator to suppress, as much as possible, the photon flux coming from the walls and not directly from the plasma core, thus improving the signal-over-noise ratio. The tilted conical holes have a lower diameter of 40 mm and are collinear in pairs, pointing to the centre of the plasma chamber. The magnetic trap is surrounded by an iron yoke (in blue), also drilled with 12 holes (of diameter 88 mm) along the cone of view of the 12 collimators. At the end of each collimator, a quartz window of thickness of 3 mm and diameter 88 mm (in yellow) was placed.

Fig. 17

a Sketch of the plasma chamber surrounded by superconducting coils (yellow) and hexapole (red). The magnetic system is surrounded by the cryostat structure (pink) in which tapered holes used as lines of sight for γ-ray detection are shown. b Ray-tracing simulations for $\gamma$-rays

Since the harsh environment (the noise is represented by the intense plasma self-emission) strongly affects the signal-to-noise ratio, HPGe (70$\%$ of relative efficiency $\epsilon _r$) detectors were chosen for the high resolution (0.2$\%$ @ 1MeV) [67]. Due to technical limitation in the number of holes (cones of view that can be created) in the magnetic trap, the best compromise was obtained using 14 HPGe detectors (2 axially and 12 radially) surrounding the magnetic trap and placed collinearly at each collimator. The total photopeak efficiency of the array (including the geometrical acceptance) was estimated through GEANT4 simulations due to the finite size of the emitting source. Simulations were performed considering an isotropic ellipsoidal source placed around the centre of the plasma chamber, having semi-axes lengths of 79, 79 and 56 mm, respectively, for the x, y and z axis (corresponding to the plasma volume and shape provided by the magnetic field profiles in the PANDORA plasma trap). The $\gamma$-ray energy range explored extends from 40 keV to 2 MeV. An example of ray-tracing simulations for $\gamma$-rays in the 100 keV–1 MeV range is sketched in Fig. 17b. The trend of the array efficiency as a function of the incident $\gamma$-ray energy is shown in Fig. 18, its value ranges between 0.1 and 0.2$\%$ and depends on the energy of the detected $\gamma$-ray.

Fig. 18

Efficiency vs. $\gamma$-ray incident energy for the HPGe detectors

6.2 Virtual experimental run for the $^{{\textbf {134}}}$Cs isotope

In order to estimate the feasibility of the PANDORA experiment, simulations of a “virtual experimental run” were performed for the three first physical cases. The results demonstrated that experimental runs durations could take from several days to 3 months, depending on the isotope under investigation, to reach $3\sigma$ significance level. The results for the $^{134}$Cs case (which the known lifetime for the neutral state is unsatisfactory, as it inhibits the proper reproduction of the two s-only isotopes $^{134}$Ba, $^{136}$Ba in suitable proportions [5, 6]) are summarized in Fig. 19 [68]. Hence, a measurement in plasma could shed light on the role played by the ionization state affecting the $\beta -$decay.

Fig. 19

Confidence levels that it is possible to obtain after a given measurement time, reported on the x axis, vs. the decay rate (or the lifetime) of the Caesium radionuclide

The green vertical axis reports on the lifetime expressed in years, starting from the lifetime of the neutral isotope (2.97 years) to the values of lifetimes predicted by the theory [2] and that are feasible in the PANDORA trap (the expected collapse of the lifetime as theoretically evaluated is up to 2 order of magnitude at around 20 keV, no variations are instead expected for values lower than 5 keV, as also shown in Fig. 1, right). Considering these lifetimes, the effective activity in the plasma (expressed in cps)—assuming a plasma volume of 1500 cm$^3$ with a concentration of 0.00001% of Cs with respect to the buffer density (10$^{13}$ ions/cm$^3$) - are shown in the blue vertical axis. Finally, taking into account the efficiency estimated by GEANT4 simulation, the cps detected in the multi detectors array were obtained (black vertical axis). Along the x axis, the measurement time is reported. The error over the background (mainly represented by the intense plasma self-emission) is estimated as the square-root of counts in the energy resolution window, whilst the counts due to the real decays occurring in the plasma are summed-up linearly with time according to Eq. 1. Pseudo-colours give the total number of counts of $\gamma$-ray at the peak of interest (@ 795.86 keV). Black dashed lines represent iso-significance contours giving $\sigma$-confidence-level at each given combination of expected activity versus measurement time. The black region is such that the significance is worse than 1$\sigma$. This plot summarizes the feasibility of the measurement and shows that for the variations of the lifetime expected from the theory in our laboratory plasmas (assuming, for example, a variation of an order of magnitude (rate 0.142)) a $t_m\sim$ 7 h is needed to obtain a $3\sigma$ level of confidence. Similar plots were performed for the other physics cases [65]. Future PANDORA measurements will allow to discriminate between different sets of theoretical predictions of the expected variation of the lifetime (i.e. [69]) and to eventually better fit the data of nucleosynthesis which disagrees with the current decay rate predictions (i.e. [14]).

7 Conclusion and perspectives

The experimental study here presented was aimed to investigate ECR plasmas proprieties, in stationary versus turbulent regimes, by means of a multi-diagnostics system (Sect. 2) and new analysis techniques (Sect. 3). The experimental results demonstrated that the plasma diagnostic plays a crucial role in the new scientific scenario where laboratory ECR plasmas can become new environments for fundamental nuclear physics investigations (Sect. 1).

The plasma turbulence regimes were studied for the first time in a quantitative way, by introducing the semiempirical parameter $I_S$ as an indication of the plasma instability strength (Sect. 3.2). These studies allowed to demonstrate the damping of the instability under the TCFH plasma heating mode (Sect. 4.1), which causes a reduction of $I_S$ and an improving of the plasma confinement (Sect. 4.2). This result has a twice relevant outcome: (i) a new “knob” for mastering the plasma instabilities and improving the plasma confinement, very useful for PANDORA to maintain the plasma stable for weeks; (ii) the mechanism of plasma instability is scientifically relevant by itself since there is the opportunity to reproduce and study phenomena of astrophysics interest (such as the Cyclotron Maser Instability, which is a typical kinetic turbulence occurring in astrophysical objects [70,71,72]) in the laboratory plasmas, measuring several properties even in the transient regimes.

The studies about plasma confinement and dynamics of losses were performed by means of X-ray imaging analysis, correlating the results with the plasma instabilities strength (Sect. 4). The integrated X-ray imaging allowed to investigate the local energy content of the plasma and the energy released as bremsstrahlung radiation caused by fluxes of deconfined electrons impinging on the plasma chamber walls (Sect. 4.2). The advanced analytical methods developed for post-processing SPhC images (Sect. 3.1) allowed powerful investigations in terms of space-resolved spectroscopy and imaging (Sect. 5), putting in evidence the local displacement of electrons at different energies, as well as of plasma ions, highlighted by fluorescence lines emission. It allowed to perform the imaging of the elemental distribution, distinguishing the emission coming from plasma only, versus the ones coming for the plasma chamber walls materials, as reported in Sect. 5.1. A complementary analysis was carried out not by energy-filtering of the images but rather making a local energy resolved analysis of the emitted X radiation (Sect. 5.2). A proper combination of these techniques makes possible the local plasma parameters monitoring (i.e. local evaluation of plasma density and temperature). SPhC images also allowed to investigate the plasma structure changes and the plasma radius. In perspective, by using two CCD pin-hole camera setups simultaneously (along the axial line and the radial one) it will be possible to estimate the plasma volume. Its determination will be very useful for PANDORA since the activity of in-plasma isotopes directly depends on plasma volume (Eq. 1).

Finally, GEANT4 numerical simulations (Sect. 6) were finalized to design the $\gamma$-ray HPGe detector array (Sect. 6.1) and to check the sensitivity of the PANDORA setup in a kind of virtual experiment, estimating the experimental run durations for getting enough statistical significance in terms of $\sigma$-levels (Sect. 6.2). The measurement time needed to reach at least 3-$\sigma$ level of confidence is expected to range from few hours (for $^{134}$Cs) to 70–80 days (for $^{176}$Lu, which is the most challenging case in PANDORA) assuming the theoretically expected variation in the lifetime of the radionuclides in our laboratory plasmas (at around 10 keV of electron temperature).

In conclusion, the simultaneous use of tools and methods here presented will guarantee to suitably operate following the experimental procedure of PANDORA: by mastering plasma stability even for weeks, by detecting through the HPGe detectors array the $\gamma$-rays emitted after the $\beta$-decay and by correlating the in-plasma radioactivity directly to the plasma density and temperature, monitored by the multi-diagnostics also in a space-resolved way.