Dynamics of laser-ablated molybdenum plasma in vacuum: a novel spectral matching algorithm based on Saha–Boltzmann equilibrium for n_e and T_e determination in fusion wall diagnostics

Xiaohan Hu, Huace Wu, Ding Wu*, Xinyue Wang, Shiming Liu, Ke Xu, Ran Hai, Cong Li, Chunlei Feng and Hongbin Ding*
Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, School of Physics, Dalian, 116024, China. E-mail: dingwu@dlut.edu.cn; hding@dlut.edu.cn

First published on 8th July 2025

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1 Introduction

During the operation of a tokamak device, especially when a large number of energetic particles in a high-temperature plasma directly bombard the plasma-facing materials (PFMs), plasma–wall material interaction (PWI) occurs inevitably.^1,2 The PWI process in the tokamak leads to the weakening of the wall, the easier formation of microcracks, and tritium retention, which seriously affect the confinement performance and stability of the plasma.^3–7 This is one of the key issues for the success of magnetic confinement fusion devices.^8,9 Molybdenum (Mo) has been used as a PFM in EAST due to its excellent characteristics of high melting point, low sputtering rate, low fuel retention, and low cost.¹⁰ Diagnostics of pure Mo plasmas in vacuum can provide essential baseline data free from impurity interference for PWI models while enabling the mapping of material migration and redeposition. However, conclusions drawn from such studies must be cautiously extrapolated to fusion device environments, requiring a gradual approach with progressive experimental steps—such as introducing impurity gases—to progressively approximate actual PWI scenarios.^11,12

Laser-induced breakdown spectroscopy (LIBS) is a non-contact, in situ, online and remote elemental analysis technique that has been proposed and demonstrated as a method for in situ wall element diagnostics in fusion devices.^13–16 However, the pulsed laser ablation processes are highly complex, involving physical phenomena such as optical absorption, heat transfer, phase transition, plasma dynamics and interactions with the surrounding environment, which lead to significant challenges in the accurate quantitative analysis of LIBS.¹⁷ Therefore, spatio-temporal resolution of LIBS is crucial for studying the emission intensity of elements, plasma dynamics evolution, and further understanding the characteristics of laser-induced plasma.¹⁸ Although achieving true volumetric spatial resolution of the plasma is challenging, spatial resolution can be approximated by collecting integrated emission spectra at different distances from the target surface. In recent years, many studies have been reported on the spatio-temporal evolution of LIBS spectra through theoretical simulations and experimental methods.^19–22 Fast imaging using an ICCD also contributed to understanding the spatio-temporal evolution of laser-induced plasma.^23–25

The parameters of plasma such as electron temperature (T_e) and electron density (n_e) of laser-induced plasma exhibit spatio-temporal non-uniformity. In the relatively high-temperature core region of the plasma, there are more particles in highly excited states; in the relatively low-temperature edge region, the number of particles in highly excited states is comparatively fewer.²⁶ This non-uniformity leads to reduced spectral line intensity and increased full width at half maximum (FWHM) of the spectral lines, which greatly reduces the accuracy of quantitative analysis.²⁷ Therefore, understanding the spatial and temporal distribution of T_e and n_e is important for accurate quantitative analysis of LIBS. Previous studies have calculated the temporal evolution of T_e and n_e in laser-induced plasma using the Boltzmann slope method and Stark broadening method.^20,28–30 But there are still problems such as missing electron collision parameters for calculating electron density, or spectral line overlap that cannot be decoupled, biases introduced by the manual selection of spectral lines, and complicated calculation process. Additionally, other methods have been explored for calculating the T_e and n_e of laser-induced plasma. For instance, Oliver et al.³¹ combined Bayesian inference with physical models to automatically estimate T_e and n_e using emission spectra. Bredice et al.³² proposed a three-dimensional Boltzmann plot method, which uses the independent components of the LIBS spectra with given temporal evolutions to obtain the temporal evolution of T_e. Liu et al.³³ determined the spatio-temporal evolution of T_e and n_e from the Stark broadening and shift of the laser-induced plasma atomic spectral lines in vacuum. However, systematic studies on the spatio-temporal evolution of T_e and n_e in vacuum are still limited.

In recent years, the spectrum matching algorithm (SMA) is a promising method that has been widely used in many fields. Zhang et al.³⁴ firstly combined the SMA based on four methods with LIBS technology. By comparing the spectral similarity between tablets and simulated samples, this work established a mineral pigment LIBS database to enable the rapid identification of mineral pigments in Dunhuang murals. He et al.³⁵ used waveform and spectral absorption indices as classification features to identify poppy plants by hyperspectral imaging and spectral matching classification technology. This method may be feasible for calculating plasma parameters such as T_e and n_e.

Calibration-free LIBS (CF-LIBS) is a famous element quantitative analysis method without standards, proposed by Ciucci et al. in 1999.³⁶ In tokamak fusion devices, CF-LIBS serves as a diagnostic tool for monitoring fuel retention (e.g., hydrogen isotopes) and impurity deposition (e.g., tungsten, carbon) on PFMs, providing essential experimental data support for the investigation of PWI mechanisms.^37–40 T_e governs the population distribution of excited states through LTE, directly dictating the correlation between spectral line intensities and elemental concentrations, and n_e influences ionization equilibria (as described by the Saha equation) and Stark broadening of spectral lines in CF-LIBS. Consequently, determination of the T_e and n_e is a critical step for CF-LIBS analysis, enhancing accuracy of quantitative analysis and correcting self-absorption effects.^26,41,42

Based on the assumption of local thermodynamic equilibrium, this work fits the corresponding spectral lines in the laser-induced Mo plasma spectral database using the Saha ionization equilibrium equation and the Boltzmann distribution equation. The similarity between the simulated spectra and the processed experimental spectra is compared by SMA, resulting in the determination of T_e and n_e. Furthermore, the spatio-temporal evolution of Mo T_e and n_e in vacuum are further analyzed. This work greatly simplifies the calculation process of T_e and n_e, provides more comprehensive data support for the establishment of theoretical models of laser-induced plasma in vacuum,^43–46 and facilitates the rapid and accurate determination of characteristic parameters of laser-induced plasma to enhance the precision of quantitative analysis in CF-LIBS.

2 Experimental setup

The schematic diagram of laser-induced Mo plasma spectra diagnostic experiment is shown in Fig. 1. This system mainly consists of a laser, a spectrometer, a vacuum chamber, and optical components.

A nanosecond pulsed Q-switched Nd:YAG laser (Brilliant Eazy, Quantel) with 5 ns pulse width and 2 Hz repetition operating at its fundamental frequency (wavelength of 1064 nm) was used to produce plasma. The laser beam was focused on the pure Mo sample surface through a plano-convex lens (focal length of 500 mm, diameter of 100 mm). The laser energy reaching the sample surface was 150 mJ. The laser spot diameter (measured by scanning the ablation crater) was approximately 0.8 mm, and the laser power density was 6.0 GW cm⁻².

The pure Mo sample was fixed on a two-dimensional stepper motor moving platform in the vacuum chamber to conveniently change the ablation position. The vacuum chamber was made of stainless steel with a diameter of approximately 35 cm and three quartz glass windows with the diameter of 15 cm were mounted on it. The internal pressure of the vacuum chamber could be reduced to 3 × 10⁻⁵ mbar using a molecular pump (KYKY, FF-200/1200E, China) coupled with a mechanical pump (KYKY, RVP-8, China) to simulate the vacuum environment of the tokamak.

The laser-induced plasma was characterized using optical emission spectroscopy and fast imaging methods. The spectral signal was collected by a plano-convex lens (focal length of 500 mm, diameter of 100 mm) perpendicular to the laser beam. The plasma optical emission was guided into an optical fiber bundle, which was composed of 20 linear array fibers with the core diameter of 350 μm and an array length of 7 mm at both ends. As shown in Fig. 1, the ratio of the collecting lens object distance (26 cm) to image distance (17 cm) was approximately 1.5, and the plasma collection region was 10.7 mm. This setup allowed for the simultaneous recording of optical emission from 20 positions in the plasma. The other end of the fiber bundle was coupled with an Andor spectrometer equipped with the ICCD camera (ANDOR iStar, 340T). The ICCD detector was 1024 × 1024 pixels. The spectrometer (Shamrock SR-750-R) grating was 1200 lines per mm and the wavelength coverage was approximately 13 nm under the experimental setup. The spectral line distribution and spatio-temporal evolution of the laser-induced Mo plasma are investigated in this method, and the experimental results obtained are presented in Sections 3.1 and 3.2, respectively.

To investigate the optical emission corresponding to the spectral range of 425 nm to 435 nm, another ICCD camera (Andor iStar, 340T) of the same model equipped with a bandpass filter (425–435 nm, Thorlabs) was utilized to take transient plasma images perpendicular to the laser direction. A digital delay generator (DG645) was employed to control the delay time between the laser pulse and the ICCD detector gate.

Fig. 2 shows the time synchronization of the experiment. The timing of two ICCD cameras was kept consistent. In the experiment, we set the zero time t₀ to about 5 ns before the peak of the laser pulse, t_d was the gate delay time and t_w was the gate width of the ICCD. To compensate for the intensity decay due to plasma expansion, t_w was set to values related to the gate delay time, t_w = 0.3 ns + 0.1 × t_d. All spectral intensities shown in this paper were normalized by the gate width. To avoid the interference of the sample surface impurity elements and roughness on the experimental results, the sample was pre-cleaned with 10 laser pulses before collecting data at each new ablation point. Each spectrum was the result of averaging the spectra obtained from ten laser pulses.

3 Results and discussion

3.1 Time-resolved optical emission spectroscopy

The spatially integrated spectra of laser-induced Mo plasma at different delay times are shown in Fig. 3. Roman numerals I and II represent the spectral lines of Mo atoms and ions, respectively. It can be seen in Fig. 3(a) that the continuum radiation occupies a dominant position at the beginning of plasma formation, which then rapidly decreases. When the time grows to 60 ns, the ionic spectral lines appear; at 210 ns, both the atomic and ionic spectral lines are more obvious; and at 450 ns, the ionic spectral lines have almost disappeared, while the atomic spectral lines are still present.

Table 1 presents some significant spectral lines and related parameters observed in Fig. 3. A total of 7 atomic lines and 6 ionic lines are listed. These parameters can be found in the Kurucz database.⁴⁷ Some spectral lines in the Mo plasma spectrum overlap due to spectral resolution limitations. In order to facilitate the analysis, the continuum background of Mo plasma at 435.51 nm, the singly ionized line Mo II at 436.37 nm and the atomic line Mo I at 428.86 nm are selected as the subjects of analysis.

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Fig. 4(a) shows the temporal evolution of the spatially integrated intensity of laser-induced Mo plasma under a vacuum condition of 3 × 10⁻⁵ mbar, where the intensity of the continuum background is reduced to one-twentieth and the intensity of the atomic spectral line is multiplied by a factor of ten for better comparison. During the initial radiation phase of Mo plasma, the bremsstrahlung intensity is very high due to the extremely high plasma temperature, resulting in a very strong continuum background signal that peaks at around 20 ns. As the plasma gradually expands and cools down, the plasma temperature and particle number density become lower, the bremsstrahlung radiation decreases rapidly. The intensity of the continuum background signal decreases sharply with a very short duration, and basically stays at a very low level after 100 ns. The intensity of the ionic spectral line Mo II reaches the peak at about 100 ns and then decreases rapidly, almost disappearing after 300 ns. The peak of the atomic spectral line Mo I appears later, which is about 200 ns, then decreases slowly with time and lasts for a long lifetime. This is due to the fact that Mo ions are in a higher ionization state compared to excited Mo atoms, which decay more rapidly, and the atoms can be produced through ion recombination and thus have a longer existence time. According to the temporal evolution profile of the plasma emission shown in Fig. 4(a), approximate time scales of nanosecond laser-induced plasma dynamics according to the optical emission are given in Fig. 4(b), including laser–plasma interactions (LPIs), continuum radiation, ionic and atomic optical emission. Although the formation of multiply charged Mo ions is expected under high laser fluence in vacuum,⁴⁸ their characteristic emission lines are not observed in the present spectral measurements. The prominent spectral lines of highly charged molybdenum ions (e.g., Mo⁴⁺–Mo⁷⁺) are primarily located in the vacuum ultraviolet (VUV), deep ultraviolet (DUV) and extreme ultraviolet (EUV) spectral regions,⁴⁹ which are beyond the detection range of our current spectrometer (425–435 nm). Moreover, even if several lines of these ions fall within the measured range, they are likely to be obscured by the much stronger emission from neutral, singly ionized Mo lines and continuum radiation noise. Although these lines were not captured in our current setup due to instrumental limitations, they deserve further attention in future studies with extended spectral coverage.

3.2 Fast imaging of laser-induced plasma and space-resolved optical emission spectroscopy

To comprehensively understand the dynamic evolution of laser-induced Mo plasma, we have investigated the fast imaging of laser-induced plasma and conducted spatially resolved emission spectroscopic analysis of the plasma in vacuum. Fig. 5 shows the two-dimensional images of the plasma plume recorded by an ICCD camera at different delay times under the pressure of 3 × 10⁻⁵ mbar. The images are integrated using a band-pass filter to capture all visible emission signals in the wavelength range of 425 nm to 435 nm. The intensity of every image is normalized with its respective maximum intensity value for clear viewing.

Fig. 5 shows that under the vacuum condition of 3 × 10⁻⁵ mbar, the plasma expands rapidly with a short duration and the emission intensity becomes extremely weak after 1 μs due to the absence of the confining effect from the background gas, which is in agreement with previous related studies.^23,50 The plasma plume expands freely and adiabatically in a cylindrical shape in accordance with the theoretical simulation predictions of laser-induced plasma dynamics evolution in vacuum.^51,52 Assuming that there is no spatial variation in the plasma temperature, the plasma expansion state can be described by the following equation:⁵²

	(1)

In this equation, X₀, Y₀ and Z₀ represent the initial scales of the plasma in each direction when the laser pulse ends. The constant adiabatic exponent γ = C_p/C_V is the ratio of heat capacities at constant pressure and volume. β(γ) is a function related to γ, which is proportional to the initial energy of the plasma and inversely proportional to the particle mass. The equation above indicates that the acceleration of plasma species depends on their initial energy, size, and mass. This model is applicable to all species including atoms, ions, molecules, neutral particles, small clusters and so on. Due to their different masses, different species exhibit varying rates of expansion. The initial transverse scale of plasma Mo (X₀ or Y₀ ∼ 0.4 mm) can be approximated by the radius of the laser spot, while the initial height in the vertical direction (Z₀ ∼ 20 μm) is approximately expressed as Z₀ ≈ 0.01 × [T/A]^1/2 cm, where T represents the plasma temperature and A represents the atomic mass number.⁵¹ From the equation, it can be observed that velocities in each direction are determined by the length of that direction. Since the initial transverse scale of the plasma is much larger than the vertical scale, it accelerates faster in the vertical direction. As the plasma expands, most of the energy is converted to particle kinetic energy, and no additional energy is available for the expansion process. Therefore, the plasma becomes longer in the vertical dimension, forming the characteristic plasma shape that extends rapidly in the vertical direction from the surface of the target material as shown in Fig. 5. It is noted that the plasma expansion exhibits slight asymmetry, which is most likely caused by the fact that our laser beam profile is not a perfectly symmetric Gaussian spot.

Spatial distribution of the continuum radiation, various ionic and atomic spectral lines in the Mo plasma at different delay times can be observed in Fig. 6. The wavelength range of the spectra is from 423.5 nm to 436.8 nm, and the vertical axis represents the distance from the sample surface. In the early stage (20 ns), the plasma spectral emission is primarily composed of continuum radiation, which only appears near the target surface (<3 mm). The Mo singly ionized spectral lines appear before the Mo atomic spectral lines and expand rapidly over time, reaching up to approximately 8 mm from the target surface. The Mo atomic spectral lines appear later, reaching up to about 10 mm from the target surface, with a spatial expansion scale larger. At 210 ns, it can be seen that during the plasma expansion process, atoms and singly ionized ions of Mo are spatially separated.

The positional changes and distribution features of singly ionized ions and atoms in the Mo plasma with time can be better understood through Fig. 7. In Fig. 7(a), it can be clearly seen that the signal of Mo II 436.37 nm spectral line is generated on the sample surface at about 50 ns and expands outward with time, peaking at 80 ns in a distance of 0.5 mm from the target. The signal basically disappears after about 200 ns, and the diffusion range is up to about 4.0 mm. Fig. 7(b) shows that the signal of Mo I 428.86 nm spectral line is generated on the sample surface at about 100 ns, which is later than the generation time of Mo II 436.37 nm spectral line. The signal peaks at 200 ns in a distance of 1 mm from the target and disappears after 500 ns, the diffusion range is about 5.6 mm. Linear fitting of the diffusion velocities of Mo atoms and Mo ions is performed in Fig. 7, and the diffusion velocities of Mo ions and atoms are found to be 2.52 × 10⁴ m s⁻¹ and 1.49 × 10⁴ m s⁻¹, respectively. The velocity of ions is larger than the velocity of atoms. In the expansion of the Mo plasma, the average thermal kinetic energy acquired by electrons and ions is basically equal. However, as the electron mass is much smaller than the ion mass, the electron velocity is much greater than the ion velocity. Electrons escape from the surface of the target material earlier than ions, leading to spatial charge separation of electron and ion flows and creating a negative potential, forming a “transient sheath”.¹⁹ This negative potential in turn prevents the diffusion of the electron flow and accelerates the ion flow, resulting in the Mo ion velocity being greater than the Mo neutral atom velocity. As seen in Fig. 7(c), the fitting speed of the Mo plasma images is 2.10 × 10⁴ m s⁻¹, which is between the velocities of Mo ions and atoms. That is, the luminescence of Mo ions and atoms cannot be decoupled in the plasma images under vacuum conditions.

3.3 Spatio-temporal evolution of electron temperature and electron density

In the laser-induced plasma, atoms or ions transition from higher energy levels to lower energy levels, emitting photons during the transition process and forming characteristic spectral lines of certain wavelengths. Assuming the plasma is in a state of local thermodynamic equilibrium, the number density of particles at each energy level follows the Boltzmann distribution. The number density of particles at a given energy level i is n_i:

	(2)

where n_a is the total number density of the particles being studied, g_i is the degeneracy of the energy level i, E_i is the excitation energy of the energy level i, k is the Boltzmann constant (approximately 1.38 × 10⁻²³ J K⁻¹), T_e is the electron temperature, and U_a is the partition function corresponding to the electron temperature. The partition function is expressed in the following table, and the parameters (e.g., atomic energy level, degeneracy) in the expression are from NIST:⁴⁹

	(3)

The intensity of a spectral line in the plasma corresponding to a transition from energy level i to energy level j can be expressed as:

	(4)

In the formula, A_ki is the transition probability for a particle transitioning of energy level i to energy level k, which is the rate or probability of the transition per unit time. From this equation, it can be seen that the spectral intensity is correlated with the electron temperature. The ratio of ions with different charges satisfies the Saha ionization equilibrium equation:

	(5)

where n^z is the number density of particles with charge state z, n^z−1 is the number density of particles with charge state z − 1, n_e is the number density of free electrons, U_z and U_z−1 are the partition functions of particles with charge states z and z − 1, m_e is the mass of the electron, k is the Boltzmann constant, ℏ is the Planck constant, ΔE_i is the ionization energy of a particle transitioning from charge state z − 1 to charge state z, and ΔE is the ionization potential depression factor due to plasma Debye shielding. The expression is given by eqn (6), with a magnitude of approximately 0.1 eV.

	(6)

From this, it can be seen that the emission intensity of the spectral line of atoms and ions in Mo plasma is correlated with the electron temperature and electron density. All spectral lines in the spectra are considered to have the same broadened Voigt profile, which is obtained by convolving fixed Gaussian broadening (ω_G = 0.062 nm, instrumental broadening measured experimentally) and Lorentzian broadening (ω_L, fitting value). Based on the aforementioned theory, we can obtain detailed profile of each spectral line for any chosen combination of n_e and T_e to constitute the simulated plasma spectra.

Using the spectral matching algorithm (SMA), we iteratively compare the simulated spectra with the experimental spectra to determine the plasma parameters. The degree of similarity between the two spectra is quantified using the correlation coefficient, which evaluates the strength and direction of their linear relationship. The definition of the correlation coefficient is:

	(7)

where cov(X,Y) is the covariance of X and Y, var(X) is the variance of X, and var(Y) is the variance of Y:

Cov(X,Y) = E[(X − E(X))(Y − E(Y))] = E(XY) − E(X)E(Y)	(8)

Var(X) = E((X − E(X))2) = E(X2) − E2(X)	(9)

In this work, the fitting and matching process is implemented using MATLAB 2021 software. In this algorithm, within the spectral wavelength range of 423.6 nm to 436.7 nm, we utilized a total of 136 Mo atomic lines and 7 Mo singly ionized ion lines which can be found in the database of Kurucz.⁴⁷ Due to the large number of spectral lines and their easy accessibility in the database of Kurucz, they are not listed individually. The experimental spectra are preprocessed, including baseline correction, noise reduction, and wavelength calibration. During the fitting process, the electron temperature, electron density, and Lorentzian broadening are iteratively adjusted to maximize the correlation coefficient between the simulated spectrum and the processed experimental spectrum. The optimal correlation coefficient is compared to a predefined threshold (set to 0.9) to evaluate the reliability of the fit. If the maximum coefficient exceeds the threshold, the fitting result is considered valid. It is important to emphasize that the threshold functions solely as a post-fitting quality criterion and does not influence the fitting process itself.

In conventional Boltzmann plot methods, line overlap can cause the total intensity of overlapping peaks to be incorrectly assigned to individual transitions, and reduce matching accuracy between the simulated and experimental spectra in the overlapped regions. Hence, a careful selection of isolated spectral lines is required. This involves extracting line intensities, identifying non-overlapping lines, and often performing line-by-line fitting, which becomes extremely laborious and error-prone when the spectrum is dense and contains partially or strongly overlapping features. In the SMA approach, a total of 143 spectral lines were selected with known transition probabilities and well-defined upper-level energies from the database. Importantly, the fitting procedure does not exclude overlapping lines; instead, it incorporates all relevant spectral information—including blended or closely spaced features—into a global fit. Each line is modeled with a fixed central wavelength, known transition probability, and well-characterized upper energy level. All line profiles are assumed to be Voigt profiles with the same line width. This global fitting strategy inherently accounts for line blending effects by treating the entire spectral region as a whole rather than fitting lines individually. Compared to conventional Boltzmann plot methods, the SMA method avoids the difficulty of manually selecting lines.

It should be mentioned that self-absorption is not considered here, although line self-absorption can affect the accuracy of spectral analysis by distorting line intensities and profiles, potentially leading to errors in T_e and n_e. In this work, self-absorption is not significant in vacuum (3 × 10⁻⁵ mbar), as no line flattening or reversal was observed. Moreover, the overall fitting results are consistent with physically reasonable parameters and show good agreement with expectations and prior studies (later in the text), suggesting that the influence of self-absorption is limited. It can be expected that, due to the approach of global fitting with multiple spectral lines, even if one or a few lines exhibit obvious self-absorption, the overall results would still maintain a certain degree of robustness.

Fig. 8(a) gives comparisons of the experimental spectra with the fitted spectra by the SMA method for different delay times in the distance of 0.5 mm from the Mo target surface. At the delay time of 60 ns, the determined T_e and n_e are 4.00 eV and 1.63 × 10²⁶ m⁻³ with a correlation coefficient of 0.96. At 140 ns, the determined T_e and n_e are 2.05 eV and 1.12 × 10²⁵ m⁻³ with a correlation coefficient of 0.97. At 250 ns, the determined T_e and n_e are 0.79 eV and 4.56 × 10²² m⁻³ with a correlation coefficient of 0.96. Meanwhile, Fig. 8(b) presents comparisons of the experimental spectra with the fitted spectra in the different distances with a delay time of 170 ns. In the distance of 0 mm, the determined T_e and n_e are 1.26 eV and 1.26 × 10²⁴ m⁻³ with a correlation coefficient of 0.95. At 1.6 mm, the determined T_e and n_e are 1.88 eV and 4.08 × 10²⁴ m⁻³ with a correlation coefficient of 0.98. At 4.3 mm, the determined T_e and n_e are 1.86 eV and 8.33 × 10²³ m⁻³ with a correlation coefficient of 0.98. It can be seen that the correlation coefficients of the different spatio-temporal conditions in Fig. 8 are all greater than 0.9, indicating that the spectra are well-fit. This implies that the proposed method enables rapid determination of sufficiently accurate T_e and n_e. Notably, despite achieving satisfactory fitting performance, the absence of spectral line information for certain peaks may still lead to imperfect fitting. It is anticipated that further refinement of spectral line databases will enhance the fitting accuracy of this approach in future applications.

From the spatial distribution of T_e and n_e at different delay times (Fig. 9), a much deeper understanding of the spatio-temporal evolution of T_e and n_e could be obtained. The error bars shown in Fig. 9 represent the average deviations of the fitting results from five experimental spectral data. At the same spatial position, T_e decreases rapidly from 50 ns to 210 ns, and the downward trend slows down after 210 ns. At the distance of 0.5 mm from the target surface, T_e rapidly decreases from 4.0 eV (at 60 ns) to 0.98 eV (at 210 ns), and then slowly decreases to 0.66 eV at 310 ns. T_e can be related to the dimensions of the plasma through the adiabatic thermodynamic equation:⁵²

Te[X(t)Y(t)Z(t)]γ−1 = const	(10)

where γ is the adiabatic index. In the early stage (<210 ns) the plasma thermal energy is rapidly converted to kinetic energy, resulting in an extremely high expansion velocity, and the T_e decreases rapidly with the expansion of the plasma. However, the T_e drops slowly at a lower level due to the energy gained by ion recombination. In addition, as the plasma expands and the radiation process proceeds, n_e decreases continuously and rapidly. For example, n_e decreases from 1.63 × 10²⁶ m⁻³ to 8.53 × 10²¹ m⁻³ between 60 ns and 310 ns in the distance of 0.5 mm from the target surface, and from 6.24 × 10²⁴ m⁻³ to 3.01 × 10²² m⁻³ between 95 ns and 310 ns in the distance of 2.1 mm. As reported in the literature,^53,54 the rapid decrease in n_e is caused by plasma expansion and three-body recombination. The typical process of three-body recombination can be expressed as follows:

e− + e− + X+ → X + e−	(11)

In this process, a positive ion and an electron combine to form a neutral atom or molecule, and the excess energy is carried away by another electron.

It is observed that the plasma exhibits pronounced spatial disequilibrium in T_e and n_e distributions. At 95 ns, T_e ranges from 2.69 eV to 3.23 eV, decreasing to 0.89–1.69 eV at 210 ns, while n_e shows a corresponding decline from (5.95–34.0) × 10²⁴ m⁻³ to (1.04–15.7) × 10²³ m⁻³ during the same time interval. As the plasma expands, both T_e and n_e peaks demonstrate systematic outward migration from the target surface: the T_e peak shifts from 0.5 mm (4.00 eV) at 60 ns → 1.1 mm (3.23 eV) at 95 ns → 2.1 mm (2.31 eV) at 140 ns → 3.7 mm (1.69 eV) at 210 ns, while the n_e peak evolves from 0.5 mm (1.63 × 1026 m⁻³) at 60 ns → 1.1 mm (4.96 × 1024 m⁻³) at 170 ns → 2.7 mm (1.57 × 1024 m⁻³) at 210 ns → 3.8 mm (3.83 × 1023 m⁻³) at 250 ns. The initial rapid T_e reduction (<250 ns) is attributed to radiative de-excitation processes of excited-state atoms/ions, with the system achieving spatially uniform thermal equilibrium after 250 ns. The results show that the T_e and n_e of the Mo plasma exhibit significant inhomogeneity both spatially and temporally. If this spatio-temporal evolution is ignored and the plasma is simply assumed to be uniform at a single moment or on average, it may lead to misinterpretation of line intensity ratios and ultimately introduce substantial quantitative errors. By calculating the spatio-temporal evolution of T_e and n_e, we can assess the stability of the plasma and optimize the timing of spectral acquisition, thereby enhancing the accuracy and reliability of the CF-LIBS analysis.

When applying the Saha–Boltzmann distribution to describe the plasma, the LTE condition is indeed a necessary prerequisite. A necessary (but not sufficient) criterion⁵⁵ is:

With maximum energy difference (ΔE) 2.9 eV and maximum electron temperature (T_e = 4.0 eV = 46000 K), the minimum electron density to meet the McWhirter criterion is about 8 × 10¹⁵ cm⁻³. The measured electron density is always greater than the lower limit of n_e (8 × 10²¹ m⁻³). Therefore, the plasma can reasonably be considered to satisfy the conditions of LTE.

Regarding comparisons with the existing literature in vacuum, our observed decay trends are consistent with previously reported results. For instance, Gao et al. studied the time evolution of electron density and temperature in laser-produced Cu–Al alloy plasma under vacuum, showing that at the plasma core (∼0.1 mm), the temperature decreased from 4.48 eV at 20 ns to 1.03 eV at 100 ns, while the electron density dropped from 3.97 × 10¹⁸ cm⁻³ to 4.96 × 10¹⁷ cm⁻³ over the same period.⁵⁶ In our earlier study on tungsten plasma, we observed that the electron temperature decreased from approximately 2.5 eV to 0.7 eV, and the electron density declined from ∼10¹⁹ cm⁻³ to ∼10¹⁶ cm⁻³, as the delay time increased from 100 ns to 500 ns.⁵⁷ Similarly, in a study by Harilal on laser-produced Sn plasma in vacuum, an initial electron temperature of 3.2 eV and an electron density of 7.7 × 10¹⁷ cm⁻³ were measured at a delay time of ∼100 ns.⁵⁸ Overall, the observed variations in T_e and n_e across different studies are reasonable and expected, considering the differences in target material properties, laser wavelengths, pulse durations, and energy densities employed.

4 Conclusions

This study elucidates the spatio-temporal evolution of laser-ablated Mo plasma in vacuum, offering critical insights for LIBS-based diagnostics in fusion environments. Key findings include the following. (1) Temporal sequencing: continuum radiation dominates early stages (5–60 ns), followed by ionic (50–300 ns) and atomic (80–800 ns) emissions, with neutral species persisting due to ion recombination. (2) Spatial dynamics: ionic species (Mo II) exhibit faster diffusion velocities (2.52 × 10⁴ m s⁻¹) than neutrals (1.49 × 10⁴ m s⁻¹), driven by transient sheath effects. Plasma expansion follows cylindrical adiabatic models, with vertical elongation due to energy-to-kinetic conversion. (3) Parameter evolution: rapid T_e decay (4.0 eV to <1 eV) and n_e reduction (10²⁶ to 10²¹ m⁻³) underscore the roles of expansion and three-body recombination. Spatial disequilibrium in T_e and n_e reflects non-uniform energy distribution. (4) Methodological advance: the SMA approach based on the Saha–Boltzmann distribution simplifies parameter determination, achieving high spectral correlation (>0.9) without manual line selection. These results offer a practical method for vacuum plasma diagnostics, providing direct experimental validation for theoretical models and optimizing LIBS protocols for fusion reactor monitoring. Future work will extend this framework to multi-elemental PFMs and pulsed magnetic confinement environments, further bridging laboratory insights to industrial fusion applications.

Data availability

All data supporting this article can be found within the manuscript.

Author contributions

Xiaohan Hu: methodology, experiments, data analysis, writing – original draft. Huace Wu: writing – review & editing, interpretation of results. Ding Wu: methodology, software, resources. Xinyue Wang: investigation. Shiming Liu: investigation. Ke Xu: investigation. Ran Hai: editing. Cong Li: editing. Chunlei Feng: editing. Hongbin Ding: supervision, project administration, funding acquisition, writing – review & editing. All authors participated in drafting the manuscript, discussed the results, and approved the final version.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was supported by the National MCF Energy R&D Program of China (No. 2022YFE03200100, 2019YFE03080100), and the Natural Science Foundation of China (No. 12375208, 12375203).

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