Introduction

Suprathermal electrons in plasmas is a trending topic of inertial confinement fusion and high energy density physics. Hot electrons are diagnosed with many different experimental and computational techniqes, such as x-ray emission, electron breamsstrahlung and Kα emission, spectropolarimetry and particle-in-cell1,2,3,4. Collisional radiative models with non-Maxwellian electron distribution is another alternative method to diagnose hot electrons in an emission spectra. Abdallah et al. has compared calculated and experimental emission spectra of laser produced K-shell Al plasma. The calculations have been carried out using an electron energy distribution which includes both thermal and hot electron components being the parts of a detailed collisional-radiative model. The comparison of spectra has shown that, the presence of hot electrons can alter the spectroscopic interpretation of electron density derived from standard thermal methods5.

X-pinch produced plasmas are an alternative and unique source of hot electrons. Due to the existence of a strong electric field oriented along the axis of the interelectrode gap, it is expected that hot electrons affect collisional and radiative process in these plasmas. Moreover, the very fast timescale with subnanosecond x-ray bursts coming from hot spots urges to use models far from the local thermodynamic equilibrium (LTE). Hansen et al. studied in detail that collisional excitation and ionization rates of K- and L-shell non-LTE collisional-radiative atomic kinetics models are highly sensitive to the fraction of hot electrons6.

Plasma electron temperature and density are diagnosed using the ratio of resonant to resonant or resonant to intercombination of emission lines. This procedure becomes challenging in the presence of hot electrons which can affect the individual line ratio changes in the spectra. Modeling of such plasmas requires hybrid electron distribution function in collisonal radiative models which have maxwelian and nonmaxwelian portions. So, the fraction of hot electrons, half-width half-maximum of electron beam energy standing beside plasma electron temperature, increases the dimension of the database and hence the prosecution time. Since pattern recognition techniques such as principal component analysis (PCA) and linear discriminant analysis (LDA) can reduce the dimension of a dataset while preserving its originality, they are extensively used in spectroscopy of astrophysical plasmas7,8,9.

Origin of the hot electrons generation, involved in the Langmuir turbulence, is another challenging work in high energy density plasmas10. There are many studies conducted on the complex relation between suprathermal electrons and Langmuir turbulence11. Recently, it has been shown that the interaction between electron beams, and non-linear oscillations and microturbulence can be characterized by means of predator-prey models12,13,14,15.

In this work, we have applied linear discriminant analysis (LDA) method to investigate the electron beam effects on the spectra radiated by X-pinch produced K-shell Al plasmas16. The LDA coefficients obtained are modelled by logistic growth with predator (predator-prey models) to investigate the relation between electron beams, ions and microturbulence17. In16, PCA was used to perform data analysis whereas LDA has been used to explore the hidden structures of the spectra in this paper. The main difference between PCA and LDA is that PCA considers the data as a whole, however LDA considers the differences between the classes within the data. Therefore, LDA is expected to provide more information about the details of the hidden structures. For example, it is shown that LDA extracts information about the plasma (ion and electron) oscillations and turbulence nature while PCA exracts the temperature diagnostics and collective behaivor of the plasma in the presence of the electron beams16. By reducing the dimension, spectra can be visualized in three dimension. Besides, since the eigenvectors of LDA considered in this paper are the dominant ones, they carry important information about the spectra.

The paper is organized as follows. After describing briefly the experiment in Sec. II, the third section provides the details of the non-LTE collisional radiative model and studies the electron beam effects on the K-shell Al calculated spectra applying linear discriminant analysis and predator-prey dynamics. Sec. IV discusses the modelling of experimental data, and conclusive remarks are given in Sec. V.

Experiment

The mounting is a classical one and it has been described previously16. A compact pulsed power generator is devoted to the creation of point-like x-ray sources in the keV range for radiographical application to low contrast objects. The plasma is produced by the explosion of two 25-μm Al wires on a compact L-C (40 kV, 200 kA, 200 ns) generator and the denser and brighter spots sit close to the crossing point, thus the device ensures creation of a localized and reproducible plasma. The time integrated spectrum is recorded on x-ray film through de Broglie KAP, PET or mica spectrographs installed in the equatorial plane. Due to the limited number of photons in the K-shell range, a rather wide (5 mm) entrance slit was used and it was not possible to get any spatial resolution along the X-pinch axis. Simultaneously, a set of fast detectors (filtered, absolute XUV p-i-n diodes and photoconductive diamonds) were recording the time dependence of x-ray flux in the keV region and in ns- and sub-ns regime (see Fig. 1). Time-integrated pinhole imaging in the same spectral region was performed radially on DEF x-ray film. The last two records were ensuring afterwards that there was a unique small-extension bright spot or, at least, a much brighter one, ensuring that there was no overlay of spectra coming from different sources nor geometrical blurring.

Figure 1
figure 1

Close-up of the electrical and photonic records 500 ns around the time of pinching (shot XP_630). The electrical records (voltage, B-dot probe signal, current as numerically integrated from B-dot) are normalized (−1 to 1) to their span over the whole shot (left scale). X-ray signal is figured out by the XRD signal in volt (right scale), normalized (0 to 1). Close after the x-ray emission, a rise of the inductance leads to a current dip.

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Over many tests with various wires, showing evidences of K-shell Al, L-shell Cu and Mo, the present work deals with the electron temperature and density and hot electron beam fractions of X-pinch produced K-shell Al plasma. Figure 2 presents a time-integrated x-ray spectrum of K-shell Al plasma (shot XP_630). The spectral lines of the resonant transitions of Al1 (He-like), Al2 (H-like), Al3 (He-like) and Li-like Al as well as satellite transitions of Al1, Al2 and Al3 are well resolved16.

Figure 2
figure 2

Time integrated spectrum of K-shell Al plasma (shot XP_630) with diagnostically significant lines.

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Electron Beam Effects on the Synthetic Spectra of K-Shell Aluminum

Non-LTE collisional radiative model of K-Shell Al

The details of non-LTE K-shell Al calculated spectra using collisional radiative model and the principal component analysis have been described previously16. In brief, the energy levels, collisional and radiative rates and cross-sections’ calculations were performed using the Flexible Atomic Code (FAC)18. Collisional radiative model uses the hybrid electron distribution function F(e) = (1 − f)*Fmaxwellian + f*Fnon-maxwellian to charecterize the effects of hot electrons on the spectra. Gaussian distribution with the centered energy of E0 = 10 keV was used to describe the fraction of the hot electrons. Voigt profiles of line broadening with the resolution of 300 have been used for fitting the broadening of the experimental spectra16.

Linear Discriminant Analysis

The effects of electron beams in the emission spectra can be diagnosed by the ratio of the lines which are sensitive to the high energy electron population, such as the ratio of different ion charge states of (Al1 + Al1 − IC)/Al2 and same ion charge states of (Al1 + Al1 − IC)/Al316.

Pattern recognition techniques are also alternative approaches to diagnose plasmas. Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) are the most commonly used techniques for the feature extraction and classification of high dimensional data. Recently, PCA has been applied to diagnose electron beam effects in X-pinch produced K-shell Al plasmas and the details of the extracted principal components over K-shell Al model can be found elsewhere16. However, the brief explanation of PCA, LDA and unified version of PCA and LDA are given as below.

PCA is an unsupervised method which reduces the dimension of the data but preserves its characteristics. In PCA, the vectors which have the largest variance associated to the data are computed and used as a basis for the new reduced space. Therefore, each original data is represented by its coordinate vector in the reduced space with a lower dimension.

Let Γ be a NxM matrix whose cloumns consist of the original dataset. Let μ be the mean value of the columns of Γ, and Φ be the matrix obtained from Γ by subtracting μ from the each column of Γ. The covariance matrix, which is a measure of how much variables change together, is \(\frac{1}{M}\mathop{\sum }\limits_{i=1}^{M}\,{{\Phi }}_{i}{{\Phi }}_{i}^{t}\) where \({{\Phi }}_{i}\) is the i-th column of Φ and superscript t means transposed. The vectors which has the largest variance are the eigenvectors of covariance matrix with largest eigenvalues, called principal components and denoted by |PC1>, |PC2>, |PC3>, … depending on the order of the corresponding eigenvalues.

In applications, the vector space generated by the principal compenents that correspond to the most dominant eigenvalues are considered in order to reduce the dimension.

For instance, let |PC1>, |PC2>, |PC3> be the principal components which corresponds to the largest three eigenvalues. If v is any vector of \({{\mathbb{R}}}^{{\rm{N}}}\) then the vector \(v\) can be represented by three coordinates, and in the new three dimensional vector space the coordinate vector of v is (v●|PC1>, v●|PC2>, v●|PC3>), where ● is the Euclidian dot product.

On contrary, LDA is a supervised method which also reduces the dimension of the dataset. In PCA, data is considered in its entirety whereas in LDA the focus is on the characteristics of the different classes to discriminate them. Let a dataset consists of K classes of N × 1 vectors, where each class contains M vectors. Let Γij be the i’th element of the class j for i = 1, 2, …, M and j = 1, 2, …, K. Let μj be the mean of the class j, and μ be the mean of all classes. Then the within-class scatter matrix Sw and the between-class scatter matrix Sb can respectively be expressed as:

$${S}_{w}=\frac{1}{K}{\Sigma }_{j=1}^{K}{\Sigma }_{i=1}^{M}\,({{\Gamma }}_{i}^{j}-{\mu }_{j})\,{({{\Gamma }}_{i}^{j}-{\mu }_{j})}^{t}$$
(1)
$${S}_{b}={\Sigma }_{j=1}^{K}\,({\mu }_{j}-\mu ){({\mu }_{j}-\mu )}^{t}.$$
(2)

In LDA, the eigenvectors of (Sw)−1. Sb which correspond to largest three eigenvalues are considered, and they are denoted by |LD1>, |LD2> and |LD3>. In LDA, there is a difficulty in taking the inverse of the matrix Sw if it is large. Therefore, LDA has a disadvantage in processing high dimensional data. To find a remedy for this problem in applications, the dimension of the original data is first reduced by PCA before appying LDA algorithm. In this paper, a unified PCA and LDA algorithm is applied to the spectra clustering19,20,21,22.

Spectral Representation of LD vectors

In the present work, as the first step, PCA is used to reduce the size of Sw for each electron beam fraction so that it can be inverted. The data set of any fraction consists of 230 = 5 (densities: 1 × 1019, 5 × 1019, 1 × 1020, 5 × 1020 and 1 × 1021 cm−3) × 46 (temperatures: 50, 60, …, 500) spectra of size 456 × 1. By applying PCA, the dimension is reduced to 40 by means of projecting each of the spectra into the space generated by the most dominant 40 PCs. As the second step, LDA is applied to the 5 classes of fractions where each class consists of 230 vector of size 40 × 1, and the dimension is reduced to 3. The new vector space is generated by |LD1>, |LD2> and |LD3>.

In Fig. 3, the vector spectra of ions (|LD1>) and electrons (|LD2>) are given for the beam fraction ranging from f = 0.00 to f = 0.20. Figure 3 a and b show that, the resonant transitions of Al1, Al2, Al3, Al4, Al5 and Al6 are almost absent, but their oscillations are present. Addition of the electron beams transform the evanescent waves to the standing waves23. By comparing the response of resonant transitions to the electron beams in PCA vector spectra in our recent works16, we have shown that LDA can clearly discriminate the behavior of oscillations of ions and electrons in the presence of electron beams16.

Figure 3
figure 3

Main features of the (a) ion (|LD1>) and (b) electron (|LD1>) waves with electron beam fractions f = 0.00 (none) and f = 0.05.

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Three-Dimensional Representation of LDA and predator-prey dynamics

Figure 4 shows the behavior of |LD1>, |LD2> and |LD3> coefficients with temperature increase for the fraction groups f = 0.00, 0.05, 0.10, 0.15 and 0.20. The fraction case 0.2 especially shows that the temperature increase results in plasma oscillations to move in an outward spiral turbulence.

Figure 4
figure 4

(|LD1>, |LD2>, |LD3>) points for the electron beam fractions between \(f=0.00\) and \(f=0.20\) at the electron density of ne = 1 × 1020 cm−3. The µ value has been introduced in Sec.III.b.

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Predator-Prey model

Gurcan et al. stated that the turbulence in hot dense plasma can be characterized by the predator-prey of population models14,15. The populations of two species interacting as a predator and prey can be modeled by using a pair of nonlinear, first-order equations which are the modified versions of the original Lotka–Volterra equations24. These models are called Predator-Prey models and they have been applied to many different areas such as chemical reactions25, astronomy26, economics27,28,29,30, plasmas13,15,22, and evolutionary game theory26 in order to express the complicated real life situations as differential equations.

The Lotka–Volterra equations24 are two non-linear and first-order differential equations:

$$\frac{dP}{dt}=[r-sQ]P\,{\rm{and}}\,\frac{dQ}{dt}=(\,-\,u+vP)Q$$
(3)

where \(P\) and \(Q\) are populations of prey and predator, respectively. \(r,\,s,\,u,\,v\) are parameters describing the interaction of the two species. Originally, these equations were used to exhibit the relation in a biological system of two interacting species: predator (Q) and prey (P).

Introducing a constraint on carrying capacity of the prey population forces to modify the original Lotka–Volterra equations and yields31:

$$\frac{dP}{dt}=[r(1-\frac{P}{K})-sQ]P\,{\rm{and}}\,\frac{dQ}{dt}=(-u+vP)Q$$
(4)

where \(K\,\,\)is the maximum size of preys. This is modified Lotka-Volterra model with capacity constraint. For a particular choice of parameters, the graph of P versus Q is given in Fig. 5.

Figure 5
figure 5

Scheme of a system trajectory in Predator-Prey model.

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There is a steady state at the point \(\,S=[\frac{u}{v},\frac{r}{s}(1-\frac{u}{vK})]\) and trajectory spirals around the steady state point S. There are two cases, either trajectories will spiral inwards and points converge to S or they spiral outwards and the points diverge from S.

In our case, Predator-Prey model is used to exhibit the characteristics of the plasma via the coefficients of linear discriminant analysis in Fig. 3. As a result of linear discriminant analysis for f = 0.2, it is shown that |LD2> coefficients behave like prey and |LD1> coefficients behave like a predator as in Fig. 2. This is because |LD1> coefficients deplete the |LD2> coefficients. As it is known that radiative and dielectronic recombination transitions associated to the resonant transtions are due to electron capturing process, |LD2> and |LD1> coefficients represent the ions and electrons, respectively13. Since the trajectory spiral outwards, the steady state point S is unstable. On contrary, the trajectory spiral inwards and the central point is stable in the case f = 0.0.

An important result of our work is that the point S corresponds to the lowest temperature, and the center region of the spiral has lower temperatures (50–100 eV). On the other hand, on these low temperatures, the low ionization is fixed by the electron beams13. Therefore, as the temperature decreases (|LD2> coefficient, |LD1> coefficient) which points converge to S which is an accumulation point. The temperature path is given in Fig. 6 and, |LD1> and |LD2> coefficients have inverse correlation as temperature decreases to 300 eV. |LD1> and |LD2> coefficients are accumulated above this temprature.

Figure 6
figure 6

|LD1>and |LD2> coefficient vs plasma electron temperature.

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Another key finding is that, the trend of the |LD1> and |LD2> coefficients can be predicted by using the Predator-Prey model. This estimation is more accurate than the ones in16,32 because in these papers only the information obtained by |PC1> is used whereas in this work |LD1> and |LD2> are used together for characterization, which means that less information is lost.

Since the Predator-Prey model that we are using has a carrying capacity for the |LD2> coefficients, we are also able to find a boundary for the |LD2> coefficients of the plasma.

Modelling Using PCA, LDA and Non-LTE Models

In Fig. 7, for the electron densities of ne = 1 × 1019, 1 × 1020 and 1 × 1021 cm−3, the |LD1> coefficients of the 46 × 3 spectra in the temperature range between 50 and 500 eV are given for the beam fractions of f = 0.00, 0.01 and 0.20. Figure 7 shows that |LD1> coefficients are more stabilized at f = 0.1 with the plasma electron density of 1 × 1019 cm−3. For the electron density of 1 × 1020 cm−3, |LD1> coefficients follow stabilized motion for the electron temperatures below 250 eV whereas they follow non-linear motion above that temperature. For the electron density of 1 × 1021 cm−3, |LD1> coefficients follow a nonlinear profiles for the electron temperatures below 250 eV, and they become stabilized above this temperature.

Figure 7
figure 7

|LD1> coefficients versus electron temperatures graphs for certain electron densities and beam fractions.

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There is no resemblance to both of (Al1 + Al1 − IC)/Al2 and (Al1 + Al1 − IC)/Al3 ratios16 compared to the traditional ratio diagnostics of K-shell Al spectra. This result is expected because LDA focuses on the behavior of the oscillations rather than the strong characters of resonant transitions in PCA modeling11.

The modeling of the electron temperature of K-shell Al spectrum has been done first using PCA regression modeling16. This modeling gives the plasma electron temperature of 100 eV, density of 5 × 1020 cm−3 and f = 0.20 (Fig. 8). Since PCA and line ratio diagnostics correlate well, such an agreement is expected. However, by modeling using LDA at 100 eV with density ne = 5 × 1020 cm−3, f = 0.2 and following regression curve of Fig. 7b, resonant transitions of Al1 and Al2 are not estimated well. This is due to the fact that LDA mainly focuses on oscillations of ions and electrons (Fig. 3).

Figure 8
figure 8

Experimental spectrum versus LDA and PCA model produced calculated spectrum (Te = 100 eV, ne = 5 × 1020 cm−3 and f = 0.2).

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Conclusion

The first result of our present work is that LDA can be used for the data classification of non-LTE collisional radiative K-shell Al model, and each spectrum can be characterized by the dominant LD coefficients. LDA can also be used as an alternative plasma diagnostic of K-shell Al spectra, especially for providing insights for ion and electron oscillations. However, PCA realizes a better correspondence with line ratio diagnostics for estimation of plasma parameters. Modelling of a representative K-shell Al spectrum by using PC coefficients gives Te = 100 eV, ne = 5 × 1020 cm−3 and f = 0.2 (with the beam energy centered at 10 keV). The LDA vector spectra show that the addition of the electron beam leads the oscillations of Al1, Al2 and Al3 to move as standing waves. The plot of electron temperature, |LD1>, |LD2> and |LD3> coefficients (at electron density of ne = 1 × 1020 cm−3) clearly shows that electron beam addition on the spectral model generates quantized clusters in the vector space and perform the oscillations in an outward spiral like turbulence18. Modelling of these turbulence using Predator-Prey model suggests that the center of the spiral recieves lower electron temperatures in where ionization is fixed by the electron beams. Another result of Predator-Prey modeling is that ions and electrons behave as predator and prey, respectively.