Spectroscopic Determination of Magnetic Fields in Pulsed-Power and High-Energy-Density Plasmas

Summary

This review paper surveys spectroscopic methods developed for determining magnetic fields in high-energy-density (HED) and pulsed-power plasmas, such as Z-pinches, relativistic electron diodes, and plasma opening switches. It addresses the challenges of Zeeman splitting under intense Stark and Doppler broadenings by employing polarization spectroscopy, multiplet line-shape analysis, and dopant ion dynamics. These diagnostics provide crucial spatial and temporal measurements of current distributions, flux conservation, resistivity, and plasma rotation.

Page 1 - Abstract & Introduction

IEEE TRANSACTIONS ON PLASMA SCIENCE 1 Spectroscopic Determination of Magnetic Fields in Pulsed-Power and High-Energy-Density Plasmas

Y. Maron, Fellow, IEEE, R. Doron, M. Cvejić, E. Stambulchik, D. Mikitchuk, C. Stollberg, T. Queller, E. Kroupp, G. Rosenzweig, B. Rubinstein, S. Biswas, V. Bernshtam, O. Nedostup, V. Litmanovich, V. Fisher, A. Starobinets, A. Fruchtman, Senior Member, IEEE, A. Fisher, V. Tangri, J. L. Giuliani, A. L. Velikovich, A. Dasgupta, Senior Member, IEEE, I. E. Ochs, E. J. Kolmes, M. E. Mlodik, S. Davidovits, N. J. Fisch, and M. D. Johnston

Abstract— We review spectroscopic methods developed for the determination of magnetic fields in high-energy-density (HED) plasmas. In such plasmas, the common Zeeman-splitting magnetic-field diagnostics are often impeded by various broadening mechanisms of the atomic transitions. The methods described, encompassing atomic transitions in the visible and ultraviolet spectral regions, are applied to the study of imploding plasmas (in a Z-pinch configuration) with and without pre-embedded magnetic fields, relativistic-electron focusing diodes, and plasma-opening switches. The measurements of the magnetic field in side-on observations of cylindrical-plasma configurations that are local in the radial direction despite the light integration along the chordal lines of sight are discussed. The evolution of the magnetic-field distributions obtained, together with the measurements of the plasma temperature and density, allows for studying the plasma dynamics, resistivity, and pressure and energy balance. In particular, for the Z-pinch, an intriguing question on the current flow in the imploding plasma was raised due to the observation that the current during stagnation mainly flows at relatively large radii, outside the stagnation region. For the premagnetized plasma implosions, all three components of the magnetic field (azimuthal, axial, and radial) were measured, yielding the evolution of the current flow and the efficiency of the axial field compression, as well as the relation between the geometry of the field and the plasma rotation, found to develop in this configuration. The measurements in the relativistic electron diode are used to quantify the shielding of the magnetic field by the plasmas in the diode. Also described are the experimental and theoretical investigations of a nondiffusive fast penetration of magnetic field into a low-density plasma (in the plasma-opening-switch configuration).

Index Terms— Electron and ion Diodes, line-shape analysis, magnetic-field measurements, plasma opening switch (POS), plasma spectroscopy, polarization spectroscopy, pulsed-power systems, Z-pinch.

I. INTRODUCTION THE determination of magnetic fields (B-fields) is of fundamental importance for the understanding of the operation of numerous pulsed-power and high-energy-density (HED) systems. First and foremost, knowledge of the magnetic field distribution is the only way to determine the current density distribution in the plasma [1]. Knowledge of the evolution of the magnetic-field spatial distribution is thus essential for determining the distributions of the plasma resistivity and the ohmic heating, for understanding the plasma dynamics through the j × B forces (where j is the current density vector), and for assessing the energy and pressure balance [1], [2], [3].

The two most common spectroscopic methods for the B-field determination are based on: 1) the splitting of emission or absorption lines in the plasma due to the Zeeman effect and 2) the change of an external light-beam polarization due to the Faraday rotation (the external beam is weak enough to have no effect on the plasma properties). Another spectroscopic approach is applicable for the cases of low-beta plasma (the beta parameter is the ratio of the plasma pressure to the magnetic-field pressure), where the ion acceleration is driven by a magnetic field gradient [4], [5], allowing for inferring the magnetic field distribution from the time-dependent ion velocity.

Each of the two common methods has its challenges and limitations. For the particular plasma conditions typical to HED systems, the Zeeman-splitting magnetic-field diagnostics are often impossible. The high densities and high ion velocities result in broad spectral line-shapes that smear out the Zeeman-split patterns, even when polarization techniques are employed for the suppression of the π Zeeman components of the spectrum.

Page 2 - Challenges Using the Zeeman Effect in HED Conditions

For the Faraday-rotation method, the rotation of the plane of polarization of an electromagnetic wave passing through a plasma is proportional to ∫ ne B · dl, where ne is the electron density and l is the coordinate along the line of sight (LoS) (e.g., [6], [7]). Therefore, Faraday rotation requires knowledge of the electron density at all locations along the light propagation and is also an integrated measurement over the line of view, along which usually both B and ne vary, which might lead to ambiguous interpretation of the data. Moreover, in chordal measurements in imploding plasmas, reconstructing the magnetic-field radial distribution requires assuming a cylindrical symmetry, which is usually unpredictable and is rather far from reality and thus may lead to substantial errors, see, for example, the discussion in [8]. For completeness, also mentioned is the proton beam deflectometry diagnostics (e.g., [9], [10], [11], [12], [13], [14]) that requires means for proton beam generation and integration of the effects of B and the electric field E on the proton trajectory over the entire proton path.

This review, following the presentation in the ICOPS 2021 minicourse, focuses on Zeeman-effect-based spectroscopic methods useful for HED conditions. The progress in measuring the magnetic fields in a variety of pulsed-power systems and the implication of these measurements to the understanding of the underlying physics is described. A discussion that also includes the measurements of other key parameters, namely, the electron and ion temperature and the turbulent and rotational ion motion that accompanied the B-field measurements, is given in a recent publication [15].

This review is organized as follows: it starts with a brief description of the Zeeman effect and demonstrates the difficulties in applying the common Zeeman-split diagnostics for HED conditions. Then, the use of the Zeeman effect for determining time-dependent magnetic-field distributions in pulsed-power systems, namely, Z-pinch implosions, a laser-produced plasma plume under an externally applied magnetic field, a relativistic self-magnetic-pinch diode, and magnetized plasma compression (MPC), is discussed. Also presented are B-field measurements in plasma opening switches (POSs), based on observing the ion dynamics. The measurements described are in the visible-UV band, encompassing plasma densities in the range of 10^14–10^19 cm^-3.

II. CHALLENGES USING THE ZEEMAN EFFECT IN HED CONDITIONS Consider an electric dipole transition between levels that can adequately be described in the LS approximation. The magnetic-field-induced splitting ΔE_Zeeman (both for the upper and the lower levels of the transition) in the weak-field approximation, i.e., when the perturbation due to the magnetic field is small compared to the fine-structure energy separations, is ΔE_Zeeman = g_LSJ μ_B M_J B (1) where M_J is the projection of the total angular momentum J of the given state on the direction of the magnetic field B, μ_B is the Bohr magneton, and g_LSJ is the Landé g factor, given (neglecting the relativistic corrections) by g_LSJ = [J(J + 1) + S(S + 1) - L(L + 1)] / [2J(J + 1)] (2) where S and L, respectively, are the total spin and the orbital momentum of the radiator. Note that ΔE_Zeeman ~ μ_B B ~ 10^-4 B (ΔE_Zeeman is in eV and B in T) and is independent of the transition energy. Thus, for B = 1 T, ΔE_Zeeman ~ 10^-4 eV and the induced splitting can be measured in the visible-UV region. However, for B = 10^4 T, ΔE_Zeeman ~ 1 eV and the splitting can be practically measured using X-ray transitions.

For the high plasma densities and high ion velocities in pulsed-power systems, smearing out of the Zeeman splitting due to Stark and Doppler broadenings is likely. In Fig. 1, the limitation of the Zeeman-split diagnostics due to the Stark broadening is demonstrated. The figure shows the Zeeman pattern of the C IV 3s–3p doublet transition for B = 1.5 T, assuming Doppler broadening due to Ti = 4 eV, convolved with a 2.5-Å Lorentzian that corresponds to a Stark broadening for ne ~ 5 × 10^17 cm^-3; the latter completely smears out the splitting.

Page 3 - Polarization-Based Zeeman Spectroscopy in Z-Pinches

A similar difficulty may arise due to a large Doppler broadening. Therefore, progress in the B-field measurements for HED plasmas requires extending the spectroscopic diagnostics to conditions in which the Zeeman-split pattern is not resolvable. A discussion on the application regimes and limitations of the methods that extend the Zeeman-based spectroscopy is given in [16]. We now summarize the progress in the B-field measurements using these methods and the impact of the findings on the understanding of pulsed-power systems.

III. POLARIZATION-BASED ZEEMAN SPECTROSCOPY IN Z-PINCHES In this section, we describe B-field measurements employing two methods that utilize the polarization properties of the Zeeman components. The first method is applicable for an LoS that is perpendicular to B and the second for an LoS that is parallel to B, i.e., for a cylindrical geometry requiring end-on and side-on observations, respectively (see Fig. 2).

A. Line-of-Sight Perpendicular to B For an observation perpendicular to B, the π and σ Zeeman components are of orthogonal linear polarizations. To facilitate the Zeeman-splitting analysis, researchers traditionally use polarization measurements to remove the π component. This is demonstrated in Fig. 2(a) that presents the simulation of the Al III 4s–4p doublet spectrum for B = 10 T and ne = 6 × 10^17 cm^-3, together with the shapes of the π and σ polarizations. The removal of the π polarization allows the σ components to reveal a measurable splitting. However, for a lower B-field, this technique becomes inapplicable, as demonstrated in Fig. 2(b). In this case, the polarization properties of the line emission can still be utilized to measure B by detecting the different contributions to the line shape of the π and σ Zeeman components. As clearly shown in Fig. 3(b), the orthogonally polarized spectral lines have different line widths. Since this width difference is proportional to the B-field magnitude, it can be used for the field diagnostics.

The method of the width comparison of the π and σ polarizations was used, to the best of our knowledge, for the first measurement of the B-field in an imploding Z-pinch plasma [17].

Page 4 - Analysis of B_theta and Pressure Balance in Z-Pinches

In that work, the radial distribution of the azimuthal B-field was measured as a function of time during the implosion of a CO2 puff driven by a ~300-kA, 1.6-μs current pulse. End-on observations (LoS parallel to the z-axis and perpendicular to the azimuthal magnetic field Bθ) of the plasma shell were employed. Since the linewidth difference was rather small, a reliable measurement required a high signal-to-noise ratio; thus, an averaging of the measurements for each polarization over a number of discharges was required. An example of the measured line shapes recorded in the π and σ polarizations, demonstrating the width difference used for the B-field determination, is given in Fig. 4.

The measured time-resolved magnetic-field distributions, presented in Fig. 5 [17], demonstrate the rise of B due to the compression and the rise of the circuit current. As the plasma radially implodes, the azimuthal B-field diffuses radially inward into the plasma and is compressed together with the plasma. By fitting the solution of the diffusion equation of B to the measured time-dependent radial distributions of B, the conductivity of the plasma was determined and found to be nearly in agreement with the Spitzer value that was estimated from the electron temperature and density determined spectroscopically for the same plasma [18], [19].

Using the magnetic field radial distributions obtained in this work [17], together with earlier measurements in similar experiments of the distributions of the charge states, electron density, and ion velocities [20], it became possible to determine the distribution of the ion radial acceleration due to the j × B forces. To this end, considered was a gas element residing at r0 = r(t0), where t0 is the time the gas element is encountered by the leading edge α of the imploding plasma and is ionized into singly charged ions. The radial position r(t) of this fluid element at time t is given by: r(t) = r(t0) + ∫{t0}^t v_α(r(t’), t’) dt’ (3) where v_α(r(t’), t’) is the measured radial velocity of ions of the fluid element α. The magnetic force acting on an ion in the fluid element is then: F_R(r, t) = [1 / n_α(r, t)] J_z(r, t) · B_θ(r, t) = [Z_α(t) / n_e(r, t)] J_z(r, t) · B_θ(r, t) (4) where n_α is the ion density and Z_α(t) is the time-dependent effective charge of the fluid element considered. F_R(r, t) gives the ion radial velocity V_α that would have resulted only from the j × B forces by: V_α(r(t), t) = ∫{t0}^t [F_R(r(τ), τ) / m_i] dτ (5) where m_i is the ion mass.

An example of the histories of V_α and v_α for a fluid element that is encountered by the leading edge of the imploding plasma at t = 470 ns and r = 1.35 cm is shown in Fig. 6(a). Given in Fig. 6(b) are the time-dependent total acceleration of the fluid element a_tot and the acceleration a_{j×B} due to the magnetic force. The time-dependent ion acceleration due to the thermal pressure gradient is then obtained by subtracting the contribution of the j × B forces from the total radial acceleration of the ions.

Page 5 - Line-of-Sight Parallel to B (Chordal Observations)

The measurements described above were successful only for radii > 8 mm and up to 90 ns prior to stagnation. At smaller radii or at times closer to stagnation, the Stark and Doppler broadenings were too dominant to allow for the determination of the linewidth difference of the two polarizations.

B. Line-of-Sight Parallel to B When the LoS is parallel to the magnetic field, only the σ Zeeman components are collected, and the light is circularly polarized: right-handed for σ+ and left-handed for σ−. The different polarizations of the σ Zeeman components allow each of these components to be recorded separately using a quarter-wave plate and a linear polarizer, e.g., see [22]. The wavelength separation between these two components is used for the field determination, as demonstrated in Fig. 7 and described in the studies of astrophysics and laboratory plasmas. This method is nearly unaffected by emission-line opacities since it relies on the components’ relative shift rather than on the line shapes. When applicable, this method is the most sensitive (among the Zeeman-effect-based methods) to the magnetic field [16].

In these measurements, B was obtained throughout the entire plasma shell and throughout stagnation [31], [32]. Chordal LoSs were used to observe the Zeeman effect as a function of radius. A formidable difficulty in measuring B as a function of radius over the entire imploding plasma is due to the integration of the spectra along the LoS. Generally, this means that Abel inversion of the data is required, which in imploding plasma experiments, is likely to lead to misleading results due to the common azimuthal nonuniformity of the plasma and the low signal-to-noise ratio in the data. Thus, it is essential to find ways to perform local measurements in radius, i.e., avoiding the need for Abel inversion. In addition, due to the irreproducibilities of pulsed-power experiments, determining B in various radii simultaneously in a single discharge is desirable. In the work, described in [31] and [32], both challenges were addressed. This was achieved by taking advantage of the naturally formed gradients in the plasma conditions that lead to the variation of the ionic charge state with radius, as in [20]. Measuring the field by utilizing an emission line of a certain charge state, while looking chordally at the outermost region in which the charge state is present, ensures a line of view parallel to the field and at a region limited in the radial and azimuthal dimensions, i.e., yielding B at the radius of this region (demonstrated in Fig. 8).

Page 6 - Current Redistribution and Stagnation Dynamics

Using B_θ = (μ_0 I / 2π R), where B_θ is the azimuthal field value measured at the radius R and I is the axial current within a cylinder of this radius, each such data point yields the total current flowing within the respective radius, producing a current radial distribution from which the current density can be derived. The radial distribution of the B-field throughout stagnation [31], [32] showed that the peak field remains at a radius much larger than the stagnation radius at all times. The current flowing through the stagnating plasma is found to be a small fraction of the total current. This constitutes the first direct confirmation of the conclusion, originally obtained from pressure and energy balance considerations [33], that the magnetic field pressure and energy play a small role in the plasma stagnation.

Further measurements of the radial distributions of the intensities of various charge-state transitions enabled, due to observations of various subtleties in the distribution, a simultaneous determination of the magnetic field at several radial locations throughout stagnation, and, in particular, down to small radii [40], [41]. These measurements confirmed the conclusions stated above regarding the small fraction of the current flowing in the stagnated plasma and also provided a large dataset, along the pinch column, that allowed for comparison to magnetohydrodynamics (MHD) simulations [42].

Recent detailed measurements in a small-scale Z-pinch experiment [43], employing the radially resolved B-field diagnostics that were also time-resolved, revealed a remarkable dynamics of the current. It was found that in a certain part of the pinch column, the discharge current undergoes an abrupt transition during the stagnation; from an initial stage where it is carried down to the very small radius of the stagnating plasma to a stage where most of the current flows in a low-density plasma (LDP) residing at much larger radii, while the stagnating plasma continues its implosion [45], [44].

Meanwhile, a model based on advection was then proposed [54] to explain this unintuitive bounce of the current channel and plasma motion. In this model, certain velocity profiles, satisfying the condition d/dr(|v_r|/r) < 0 (where v_r is the plasma radial velocity and r is the plasma radial position), were shown to cause a fast current transport outward, while the plasma implosion continues.

Page 7 - Multi-Species Spectral Imaging

Fig. 7 illustrates the simulation of the Al III 4s–4p doublet at B = 3 T and ne = 5 × 10^17 cm^-3, assuming an LoS parallel to B: total emission, σ+, and σ- Zeeman polarizations. Fig. 8 illustrates the schematic of measurements of the B-field radial distribution utilizing the emission lines of different charge states abundant within different radii. Fig. 9 shows the analysis of the O III 3791.26-Å and O VI 3811.35-Å lines. The measured line shapes of each of the σ polarizations are fitted with Voigt profiles, from which the Zeeman effect is obtained. The O III line was recorded from r = 8 mm and the O VI from r = 5 mm, yielding B = 7.1 ± 0.6 and 5.5 ± 0.5 T, respectively.

Fig. 10 shows a spectral image of the small-scale Z-pinch at stagnation, demonstrating the simultaneous measurement of the B-field at different radii using spectral lines from different charge states (O III and O IV). In order to accurately measure the separation between the σ Zeeman components, the two components must be recorded simultaneously on different parts of the detector, which was achieved by imaging the plasma emission with the two different polarization properties on two different regions of the detector.

Page 8 - Identifying the Outermost Radius of Line Emission

C. Identifying the Outermost Radius of a Line Emission Two conditions are crucial for the correct implementation of the method that deems Abel Inversion unnecessary, discussed in Section III-B: 1) the observed volume, identified as the outer edge of the line emission, is located at the radius associated with the chordal LoS (i.e., located at the point of the minimal distance between the chord and the pinch axis) and 2) the LoS is parallel to B in the observed volume. While these conditions are readily achieved for an azimuthally symmetric plasma column, their fulfillment in an asymmetric plasma poses a difficulty resulting from the fact that the data are chordally integrated, as clarified in Fig. 11. The figure schematically presents the cross section of a plasma column emitting an optically thin spectral line for two cases: 1) an azimuthally symmetric plasma and 2) an asymmetric plasma. Also shown are the respective chordally integrated intensity images observed from orthogonal directions.

The measurements presented in Fig. 10 for r = 1 and 3 mm were taken close to the time of stagnation and after the current redistribution. Indeed, the measured B(r) does not follow the expected 1/r decay, resulting from the current flow through the entire plasma cross section, namely, between r = 1 mm and r = 3 mm.

Page 9 - Measurements of B Having an Arbitrary Direction

In our experiments, we used simultaneously two side-on spectroscopic systems viewing the column from orthogonal directions. Repeated measurements allowed us to confirm that the outermost radius is azimuthally symmetric to within ±0.25 mm (compared to ~5-mm column radius). This verification also ensured that the deviation of the direction of view from the parallel direction to B was negligible.

Doppler shifts due to the plasma implosion provide additional information on the azimuthal symmetry of the system. In side-on observations, for an azimuthally symmetrical plasma, the LoS through the plasma edge exhibits no Doppler shift, whereas the LoS through the diameter yields the largest Doppler shifts. Fig. 12 shows a chordally integrated spectral image of the O V 2781.01-Å line along the entire diameter of the cylindrical plasma.

IV. MEASUREMENTS OF B HAVING AN ARBITRARY DIRECTION The two polarization-based methods described above require that the B-field has a dominant direction. However, when the magnetic-field direction varies in time or over spatial scales that are integrated over due to the limited diagnostic-system resolutions, polarization techniques are inapplicable. In such cases of ‘nondirectional’ fields, the only known available spectroscopic approach is that based on the comparison of line shapes of different fine-structure components of the same multiplet, suggested in [59].

The principle of this method is based on the fact that the different fine-structure components of the same atomic multiplet undergo different Zeeman splittings, while the other line-broadening mechanisms, namely, the Stark and the Doppler effects, and the instrumental broadening, are practically identical for the two components. The method is demonstrated in Fig. 13, where the Al III doublet, 4s 2S_{1/2}–4p 2P_{1/2} (5696.6 Å) and 2S_{1/2}–2P_{3/2} (5722.7 Å), is simulated for B = 8 T; with the 1/2–1/2 component being wider.

Page 10 - B-Field Measurements in Relativistic Self-Magnetic-Pinch Diodes

In high-density plasmas, the two fine-structure components may significantly overlap. In such a case, a simple comparison of the multiplet line shapes for inferring the field, as shown in Fig. 13, is not possible. Nevertheless, the diagnostics are still useful by means of detailed line shape simulations [60] of the entire multiplet, also yielding information on ne. In this approach, the Stark and Zeeman effects are assumed to be independent and the total line profile is the convolution of the Zeeman pattern and the Stark profile. Fig. 14 shows the Al III 4s–4p doublet recorded from laser-produced plasma under applied B, yielding ne = 1.3 × 10^18 cm^-3 and B = 15 T.

V. B-FIELD MEASUREMENTS IN RELATIVISTIC SELF-MAGNETIC-PINCH DIODE In high-current electron diodes, the self-generated magnetic field causes electron-beam focusing at the center of the anode [63], [64]. Plasma is formed all over the anode surface during and after the process of the beam focusing [65]. Knowledge of the beam current profile, which can be obtained from the B-field distribution, is required for understanding the pinch dynamics and for optimizing the diode performance. The magnetic field was determined in the anode plasma in the Sandia 10-MV, 200-kA electron beam diode [66], where the Zeeman effect in transitions of carbon ions present in the anode plasma was recorded, yielding the B-field in various radial positions near the anode surface.

Also, for this diode [64], the radial distribution of B in the anode plasma for different distances from the anode surface was inferred [67]. In this work, the measured B-field as a function of distance from the anode surface allowed for quantitatively determining the shielding of the magnetic field in the anode plasma. It also allowed for fitting the variation of B with the distance from the anode surface to a solution of the diffusion equation, which yielded the plasma resistivity. The latter was found to be close to the Spitzer resistivity for this plasma.

Page 11 - Z-Pinch Implosions With Pre-Embedded Axial Magnetic Field

VI. Z-PINCH IMPLOSIONS WITH PRE-EMBEDDED AXIAL MAGNETIC FIELD In a Z-pinch with a pre-embedded axial magnetic field or magnetized plasma compression (MPC), the plasma and the embedded axial field (B_z0) are compressed by the azimuthal magnetic field due to the axial current driven in the plasma. MPCs have been intensely studied during the last decade due to their importance for inertial confinement fusion (ICF), for the study of fundamental plasma physics, and for experimental astrophysics. The compression of both the plasma and axial magnetic field affects profoundly the implosion dynamics. Two such remarkable phenomena, recently discovered, are the redistribution of the axial current flow to large radii, where LDP resides [30] and self-generated plasma rotation [58].

A. Experimental Setup Fig. 17(a) shows a schematic of the experimental and diagnostic setup. A cylindrical oxygen-puff shell with initial outer and inner radii of 19 and 7 mm, respectively, and mass ~10 μg/cm prefills the A–K gap with a pre-embedded, quasistatic axial magnetic flux. B_z0 is generated by a pair of Helmholtz coils placed outside the vacuum chamber, rising in ~95 ms, to allow for the diffusion of B_z0 into the vacuum chamber and the A–K gap. Subsequently, a pulsed current (rising to 300 kA in 1.6 μs) is driven through the gas, ionizes it, and generates an azimuthal magnetic field that compresses the plasma radially inward together with the embedded B_z field.

Page 12 - Diagnostic Systems for Magnetized Plasmas

Fig. 17 displays: (a) Schematic description of the experimental setup and spectroscopic systems for the magnetized plasma diagnostics. (b) Spectroscopic system 1 used for measurements of ion velocities and plasma parameters. (c) Schematic of spectroscopic measurements of rotation and implosion velocities. (d) Spectroscopic system 2 used for magnetic field (B_θ, B_r) measurements. (e) Spectroscopic system 3 used for axial magnetic field (B_z) measurements.

B. B_θ Measurements As shown in Fig. 17(a) and (d), the imploding plasma column is observed radially. The collected light passes through a quarter-wave plate that transforms the right-handed circularly (RHC) polarized and left-handed circularly (LHC) polarized components into orthogonal linear polarizations that are subsequently split using a polarizing beam splitter. Each of the two polarizations is then imaged on a separate linear array of 50 optical fibers. The light exiting the array is imaged along the entrance slit of a high-resolution (0.3 Å) imaging spectrometer; its output coupled to a gated (10 ns) intensified charge-coupled device (ICCD).

Page 13 - Polarization Spectra and Current Escape Phenomenon

In a magnetized plasma experiment, when B_θ is measured using the method for an LoS parallel to B, one must consider that the B vector, in addition to the azimuthal component, can have axial and radial components, namely, B = B_θ θ^ + B_z z^ + B_r r^.

Fig. 18 shows a spectral image of the chordally integrated O III 3791.26-Å and O VI 3811.35-Å transitions for B_z0 = 0.26 T, showing RHC and LHC polarizations along the plasma diameter recorded on the same ICCD sensor. This allows simultaneous determination of B_θ, azimuthal rotation velocity v_θ, radial velocity v_r, electron density n_e, and electron temperature T_e.

Studying the Z-pinch implosion with B_z0 [30] revealed a phenomenon where even a weak initially applied axial magnetic field can have a significant impact on the current distribution in the plasma. During the implosion, a large part of the current is diverted from the imploding plasma to an LDP residing at large radii. Using Ampère’s law, the measured B_θ evolution at the outer plasma radius of the main imploding argon plasma shows that only ~25% of the total current flows within this plasma. It was found that the majority of the current flows at large radii through a slowly imploding LDP consisting of hydrocarbon impurities. To explain this phenomenon, a model based on the development of a force-free current flow in the peripheral LDP was suggested.

Page 14 - Axial Magnetic Field (Bz) Diagnostics

Fig. 19 shows the evolution of the current flow in the main imploding argon plasma (I_Ar ~ 50 kA) and in the slowly imploding LDP (I_LD ~ 200 kA) for B_z0 = 0.4 T.

C. Diagnostics of B_z Knowledge of the axial magnetic field (B_z) distribution and its evolution are essential for understanding governing basic processes. Presented here is a brief summary of the experimental determination of the B_z distribution near the plasma axis in oxygen-puff Z-pinch throughout the magnetized plasma implosion. In determining B_z in the plasma shell, two major difficulties are encountered: 1) the need to distinguish between axial and azimuthal magnetic fields along the LoS and 2) the absence of light emission from the on-axis region of the nearly hollow cylindrical plasma column.

To circumvent these difficulties, a dopant plasma is introduced in the central region along the imploding plasma axis using two schemes: 1) laser ablation of an Al-target attached to the anode and 2) introducing a small amount of CH4 using a jet from a central nozzle. Fig. 20 displays the spectral lines of Al III 4s–4p from the laser dopant plasma recorded at t = -115 ns and z = 7 mm for B_z0 = 0.26 T, yielding B_z = (1.6 ± 0.1) T.

Page 15 - Magnetic Field Propagation in Plasma Opening Switches (POS)

Recording the shapes of the Al III and C IV lines from dopant plasma at different times during implosion allowed for obtaining the time evolution of B_z. Fig. 21 shows that the measured values follow the curve of ideal magnetic flux compression, meaning that the axial magnetic field is ideally conserved within the imploding plasma shell in this oxygen-puff experiment.

VII. MAGNETIC-FIELD PROPAGATION IN A LOW-DENSITY PLASMA (PLASMA-OPENING SWITCH CONFIGURATION) Plasma pushing by magnetic field pressure, expected to be the dominant process in POS, was challenged by Zeeman-effect spectroscopic measurements revealing that the magnetic field penetrates into the POS plasma in a nondiffusive manner, much faster than expected by plasma resistivity. Doping the plasma using laser evaporation techniques allowed for measurements that were local in 3-D. Theories explaining fast magnetic field penetration are based on the Hall-electric-field mechanism. Fig. 22 schematically describes the accompanying ion separation, where a high charge-to-mass ratio plasma (mainly protons) is pushed ahead with velocity 2v_B, while low Z/M plasma (carbon ions) lags behind the magnetic piston.

Subsequent studies provided experimental verification of the traveling-wave-like picture of the magnetic field and the associated electric potential hill. The diagnostic approach took advantage of the POS-plasma low-beta, where the electric force accelerating unmagnetized ions is equal to the magnetic force on the entire plasma, proportional to the magnetic field gradient.

Page 16 - Electric Potential Hill Formulation & Summary

In the wave reference frame, the dopant ions climb an electrostatic potential hill, and their total energy is conserved: φ = [m_i / (2 Z_i e)] (v_B^2 - ~v_i^2) (6) where ~v_i is the ion velocity and φ is the electrostatic potential along the potential hill, v_B is the magnetic-field propagation velocity, and Z_i, m_i are the ionization degree and mass.

The plasma is highly conductive, so the accelerating electric field can be approximated by E_z = -[B / (e μ_0 n_e)] (∂B/∂z). The magnetic field is then given by: B^2(φ) = 2 e μ_0 ∫_0^φ n_e(φ’) dφ’ (7)

Employing a laser blow-off technique, a ~4-mm-wide, low-density dopant column was injected into the plasma. A submillimeter spatial resolution along the LoS was achieved by tracking the dopant-ion velocity at the center of the column. The reconstruction of the electric potential and the shape of the magnetic field front with a resolution comparable to the electron skin depth (350 μm) advanced the understanding of plasma-magnetic-field interaction in the low-beta regime.

VIII. SUMMARY The measurements of the magnetic-field distribution as a function of time and space in various pulsed power systems provide deep insight on current flow and plasma behavior, revealing unpredicted phenomena such as fast current redistribution, force-free peripheral currents, and nondiffusive Hall penetration.

Pages 16-19 - References

References [1] to [121] citing foundational and recent literature on gas-puff Z-pinches, high-energy-density plasma spectroscopy, Zeeman splitting measurements, Stark broadening models, Hall MHD simulations, electron diodes, and plasma opening switches.