Observation of fast current redistribution in an imploding plasma column

C. Stollberg,1, ∗ E. Kroupp,1 D. Mikitchuk,1 P. Sharma,1 V. Bernshtam,1 M. Cvejic,1 R. Doron,1

E. Stambulchik,1 Y. Maron,1 A. Fruchtman,2 I. E. Ochs,3 N. J. Fisch,3 and U. Shumlak4

1Weizmann Institute of Science, Herzl street 243, 7610001 Rehovot, Israel2Department of Physics, Holon Institute of Technology, Holon 58102, Israel

3Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08540, USA4Aerospace & Energetics Research Program, University of Washington, Seattle, Washington 98195, USA

(Dated: November 15, 2021)

Spectroscopic measurements of the magnetic field evolution in a Z-pinch throughout stagna-tion and with particularly high spatial resolution reveal a sudden current redistribution from thestagnating plasma (SP) to a low density plasma (LDP) at larger radii, while the SP continues toimplode. Based on the plasma parameters it is shown that the current is transferred to an increasing-conductance LDP outside the stagnation, a process likely to be induced by the increasing impedanceof the SP. Since an LDP often exists around imploding plasmas and in various pulsed-power systems,such a fast current redistribution may dramatically affect the behavior and achievable parametersin these systems.

Introduction - The force exerted on a plasma by a cur-rent and by its associated magnetic field governs variousprocesses in space and laboratory plasmas [1–16]. Thisforce is largely affected by the distribution of the cur-rent in the plasma. The implosion of a current-carryingplasma column, as in Z-pinches, is a typical examplewhere the current distribution plays a crucial role [5–7].In this configuration, the plasma is generated by a pulsedcurrent driven axially in a cylindrical gas or a metallicload. The current generates an azimuthal magnetic fieldBθ that accelerates the plasma radially inward and im-plodes together with the plasma [5–7, 9–11]. Thus, a Z-pinch constitutes a testbed for investigating the interplaybetween the current density distribution and the evolu-tion of the implosion and stagnation, found to be highlycomplex [5–7, 11, 17–19].

A detailed analysis of the energy and pressure balanceof two stagnating Z-pinch plasmas, distinctly differentin power [20], revealed that, in both the low- and high-power systems, the magnetic pressure due to Bθ has arather small contribution to the pressure balance dur-ing the stagnation. Contrary to what had been believed,at most 1/3 of the discharge current was found to flowwithin the highly compressed stagnating plasma (SP).The direct confirmation of this study requires experimen-tal measurements of Bθ during the stagnation and closeto the SP, a task that is extremely difficult due to thehigh electron density ne, the high ion velocities, and thetransient nature of the plasma [21].

A few experimental studies have measured the time-dependent magnetic field radial distribution in the im-ploding Z-pinch plasma [22–25]; those that were per-formed at the time of stagnation [24–26], outside the SPbut close enough to it, confirmed the flow of only a smallfraction of the current in this plasma. However, in thepresent work, the temporal evolution of the current den-sity radial distribution was determined down to the smallradius of the SP and, in particular, with a high spatial

resolution. This allowed for revealing a sudden transitionfrom the initial phase of stagnation where the entire cur-rent flows inside the SP to the stage where only a smallfraction of the current flows in the SP. Specifically, wemeasure that during stagnation the current transitionsout of the SP and into a low density plasma (LDP),residing at much larger radii, while the SP continuesto implode. This phenomenon might be connected tothe previous observations of low current in the SP of Z-pinches [11, 20, 24–26], and might be common to variouspulsed-power systems, including those of lower densityplasmas [11], such as plasma switches [12–14] and highpower transmission lines [15, 16].

The observation of the fast current redistribution wasenabled by performing spatially and temporally resolvedmeasurements of the magnetic-field radial distributionthroughout the entire implosion of a relatively small-scale gas-puff Z-pinch plasma. The experimental sys-tem used an initial gas column with a radial distribu-tion that provided a peak density on axis, thus allow-ing for satisfactory reproducibility and symmetry dueto the mitigation of the Magneto-Rayleigh-Taylor in-stabilities [27, 28]. The diagnostics, based on Zeeman-polarization spectroscopy, allowed for determining Bθ,even when the Zeeman-split pattern was fully obscuredby the line broadening [22–25, 29]. Also, the radial dis-tribution of charge states in the plasma was measured, asin [22, 24, 25], and was used to obtain Bθ as a function ofr from the chordal measurements without the need to in-verse Abel transform the observed line shapes (i.e. with-out the need to assume azimuthal uniformity) [24, 25].Furthermore, the detailed measurements of the plasmaparameters allowed for inferring the plasma resistivityrequired for the interpretation of the results of this study.

Experimental Methods - In the present experiment[30], an oxygen column of initial diameter ≈ 3 mm andmass of (1.6± 0.6) µg/cm (as measured by interferome-try) is injected into a 6-mm-long anode-cathode gap and

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imploded by a current pulse rising to 27 kA in 160 ns.

For the B-field measurements we use chordal lines ofsight (LOS) (Fig. 1 a). Analyzing the light from the out-ermost radius of the spectral-line emission ensures thatthe LOS is parallel to Bθ at that point, such that onlythe σ Zeeman components are observed [24, 25, 29]. Theoppositely handed, circularly polarized light of the σ+and the σ- Zeeman components is transformed into or-thogonal, linearly polarized light by a λ/4 plate and sep-arated by a polarizing beam splitter. The light is guidedthrough two polarization channels to a high-resolution(0.3�A) imaging spectrometer, chord-integrated along xand 1-D imaged along y, as illustrated in Fig. 1 a). Twoimages with a spatial resolution of ≈ 100 µm, which cor-responds to 1/5 of the SP radius, and originating fromthe exact same plasma volume, are recorded simultane-ously on different positions of a gated (typically 10 ns)intensified charged coupled device (ICCD) [Fig. 1 b)].This setup allows for the B-field determination from thewavelength separation between the σ+ and the σ- Zee-man components [21]. In order to obtain Bθ at differentradii, we use up to three different spectral lines that areradially separated.

A second imaging spectroscopic setup, providing abroader spectrum [30], is used for temperature (from theline intensity ratios [31–33]) and density (from the Starkbroadening [34–36] and absolute intensity [32]) diagnos-tics, as well as the estimation of the average charge stateZeff , which enabled us to obtain the plasma resistivity.

Results - Presented are results obtained at a distanceof z = 1 mm from the cathode valve. To demonstrate theB-field determination, Fig. 1 b) shows a typical spectralrecording of the chord-integrated plasma emission, splitinto two images with different polarizations, as explainedabove. To determine the B-field, spectral lineouts [Fig. 1c) ] are extracted at the outer edge of the O VI spectralline and fitted with Voigt-profiles. The fitting accountsfor the Zeeman components (0.1�A), the Gaussian contri-butions from the instrumental and Doppler broadenings(0.4�A) and the Lorentzian contribution (0.6�A) for theStark broadening. For the 3s - 3p transition of O VI at3811.35�A, the separation of the Zeeman σ componentsis 0.16�A for B = 1 T [21]. For the example in Fig. 1 c)we find Bθ = 2.3+0.4

−0.2 T, where the uncertainty includesthe fitting error, spectrometer aberrations, and the per-formance of the polarization optics [24, 30]. The Starkbroadening allows for determining the electron density(0.9± 0.3)× 1018 cm−3 [34, 35]. The integration regionin the radial direction is defined for each spectral lineoutindividually, depending on signal-to-noise ratio and linebroadening, and is typically 100 µm to 200µm.

To demonstrate the phenomenon of the fast currentredistribution, Fig. 2 summarizes the results obtainedfrom the B-field measurements at different times duringthe implosion (t < 140 ns) and at stagnation (t =140 ns to170 ns). For any given time, we use here the charge state

in the imploding plasma that resides at the outermostradius of the plasma, i.e. O IV at t = 50 ns and O VIlater, and show its emission-intensity radial distributionalong with the Bθ values obtained at the outer edge routof its line emission. Additionally, we show Bθ obtainedby assuming that the entire current I(t), measured bya calibrated B-dot probe (90% of I(t) are considered asthe entire current), flows within the radius of the B-fieldmeasurement, namely BB−dot(t) = 0.9 µ0 I(t)/2πrout(t).We note that each data point was obtained in a separatedischarge, where the shot-to-shot variations were signifi-cantly smaller than the error bars quoted in Fig. 2. Fort ≤ 140 ns, the data show a good agreement between themeasured (Zeeman) and the expected (B-dot) B-field val-ues, indicating that the entire discharge current flows inthe imploding plasma. At t = 170 ns, however, a signi-ficant discrepancy is observed: at most, 30% of the dis-charge current flows within the stagnating plasma thatis located at r < 1.1 mm.

At the time of stagnation, we also use O III and O IVemission lines from the LDP to obtain the current densitydistribution at r > 1.1 mm. By measuring the B-field atthe outer chords of these lines (Fig. 3), it was found thatat t = 170 ns 70% of the current flow within a radius of1.9 mm, while the entire discharge current was detectedwithin r ≈ 3.5 mm.

To demonstrate the correlation between the currentevolution and the plasma radial motion, we plot in Fig.4 the position of the imploding plasma (gray area) alongwith the positions at which the entire discharge currentwas measured (red line). Additionally, for t ≥ 170 ns,we show the positions that enclose ≈70% and ≤ 30% ofthe discharge current. It is seen that during the implo-sion (t < 140 ns), the current implodes together with themain plasma, while at stagnation (t = 140 ns to 170 ns)a current redistribution occurs to an LDP at larger radii.Remarkably, the stagnating plasma continues its inwardmotion as the current moves outwards and stays outsidethe SP for several tens of nanoseconds.

Note that applying the same analysis for locations far-ther away from the cathode, a gradually changing be-havior is found: at distances of 3 mm and 5 mm from thecathode, the discharge current is never observed to flowat small radii [30], consistent with the absence of signif-icant current in the stagnating plasma seen in previousexperiments [11, 20, 24–26].Discussion - High-spatial-resolution measurements of

the evolution of the magnetic field in a Z-pinch showed,that the entire current is carried by the imploding plasmauntil the stagnation, before a rapid transfer of the currentfrom the SP to a LDP region at the plasma peripheryoccurs, revealing a transition that has not been observedbefore.

In the following, possible mechanisms for the currenttransfer to the LDP are discussed. It was shown re-cently [37] that in ideal magnetohydrodynamics (MHD)

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FIG. 1. a) Spectroscopic setup for the determination of the magnetic field by measurement of the wavelength separationbetween the σ+ and σ- components of the Zeeman pattern (L: lens, λ/4: quarter wave plate, Pol: polarizing beam splitter, P:right angle prism). b) Typical spectral recording of the O VI 3s - 3p transition obtained at t = 100 ns (after the start of thecurrent pulse). Two spectral images of the same plasma are recorded simultaneously with different polarization properties. Thegreen line represents the unshifted spectral line, illustrating the deviation of the measured spectral lines from the unperturbedposition, where σ+ exhibits a (B-dependent) blue-shift and σ- a red-shift. c) Spectral lineouts (markers) taken from b) aty = (−1.6 ± 0.1) mm from the upper and the lower image (bright rectangles) and their fit (dashed/dotted). The solid linesrepresent the Zeeman pattern without line broadening. The separation between the profiles of 0.37 �A yields a B-field of 2.3 Tand the Stark broadening yields ne ≈ 9 × 1017 cm−3 [34, 35].

FIG. 2. Evolution of the O IV (t = 50 ns) and O VI (t ≥100 ns) emission intensity radial distribution along with themagnetic field measured at the outer chord: at t = 50 ns to140 ns, the agreement between the expected magnetic field (B-dot) and the measured magnetic field (Zeeman) indicates theconfinement of the entire discharge current in the implodingplasma. At t = 170 ns the fraction of the current that can belocated in the stagnating plasma is . 30%.

conditions, the current is expected to move radially-outward when the imploding plasma velocity profile sat-isfies ∂(|vr|/r)/∂r < 0.

Here, in addition to the effect of the plasma motion,we also consider resistance-changes induced in the LDPby the dynamics of the stagnation. Initially, the LDP is aweakly-ionized gas and its resistivity is much larger thanthe resistivity of the SP. The high current flowing inthe SP, the increase of inductance due to the decreasingradius of the current-carrying plasma approaching stag-nation, and the large resistance of the SP due to the smallradius of the plasma shell, all induce an axial electric field

in the LDP. That electric field is expected to increaseTe and, consequently, the rate of ionization and the LDPelectron density. Indeed, by the time the current is trans-ferred to the LDP, the electron density and temperaturemeasured in the LDP [30] are ne ∼ 1017 cm−3 and Te =(10± 2) eV, indicating a low resistance of this by thenfully-ionized plasma. The drop of the LDP resistanceleads to the current transfer. The partitioning of the dis-charge current I between I1, the current through the SP,and I2, the current through the LDP, has been modeledby a lumped-circuit equation d(LI1)/dt = −I1R1 +I2R2,where R1 and R2 are the resistances of the SP and theLDP, respectively, along l = 2 mm near the cathode.Also, L = (µ0l/2π) ln (r2/r1), where r1 and r2 are theradii of the SP and of the LDP that are taken as thin hol-low cylindrical plasmas. In deriving the lumped-circuitequation, the loop integration (around the circuit) in-cludes the two moving plasmas. It can be shown thatthe lumped circuit model is a discrete analog that cap-tures the physics of both the inductive current escape [37]and the resistive model in Ref. [38]. The LDP resistanceis R2 =

(me/e

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(l/A) [(keN + β)N/ne + kei], where keNand kei are the electron-neutral and electron-ion collisionrate coefficients and β is the ionization rate coefficient[39], A is the LDP cross section, N is the density of neu-trals in the LDP, and me is the electron mas. When theionization is small (N/ne � 1), the LDP resistance ishigh, dominated by electron-neutral collisions. Intenseionization, induced by a rising Te, makes N/ne . 1 anddecreases R2, which is further reduced by the rising Te,since R2 is now dominated by the electron-ion collisions.The electron density increase is determined by the rate ofionization dne/dt = βNne. This ionization can be veryabrupt, because of the exponential nature of the ioniza-

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FIG. 3. a-b) Spectral image and c-d) Fit of O IV 3p - 3dtransition at 3736.85 �A (O III 3s - 3p transition at 3759.87 �A)in the LDP at stagnation. With a separation of 0.156 �A/T(0.15 �A/T), the measurement yields a B-field of 1.7 T at r= (1.90 ± 0.35) mm (1.3 T at r = (3.55 ± 0.21) mm). Notethat the B-field value obtained for O III is a lower limit, asthe outermost radius of the transition is not accessible in themeasurement.

FIG. 4. In gray is the location of the plasma defined as theregion with more than 10% of the maximum electron densityne > 0.1 ne,max (as obtained from continuum radiation inten-sity); the light gray represents the error bar of this boundary.In red we show the positions at which a certain percentageof the discharge current was measured. At t ≥ 170 ns themeasurements were obtained in an LDP with a density lowerthan 10% of ne,max. The graph demonstrates that initially,the current implodes together with the main plasma, while atstagnation (140 ns to 170 ns), a sudden current redistributionto much larger radii takes place.

FIG. 5. Measured current partition between the implodingplasma and the LDP (markers) and its calculation by thelumped circuit model (lines). Shown are the currents dividedby the discharge current. The calculation agrees well with themeasurements.

tion equation and also because β ∼ exp(−εi/Te), εi beingthe ionization energy of neutral oxygen.

Using the measured r1(t), r2 = 2 mm, the resistanceR1(t), inferred from the measured plasma parameters[30], and assuming a linear rise-in-time from 0.5 eV att = 100 ns to 12 eV at t = 220 ns for Te in the LDP todetermine R2(t), the equations for I1 (and I2 = I − I1)together with ne(t) were solved. Fig. 5 shows the model-calculated current evolution in comparison with the mea-sured current. At a certain stage the rate of ionizationincreases dramatically and the LDP becomes fully ion-ized. The LDP resistance falls abruptly which resultsin an abrupt current transfer from the SP to the LDPbetween t = 140 ns to 170 ns. The current evolution sim-ulated agrees well with the experimental data.

Computer simulations carried out with MACH2 [40],show further that diffusive-dominated current redistribu-tion could happen even onto a strongly resistive LDP ifthe SP compresses to a sufficiently small radius, resultingin a large rise in the SP resistance. Note that in this sce-nario the current redistribution can also be interpreted asa switch of the system from an advectively-dominated toa diffusively-dominated system (due to a drastic decreaseof the magnetic diffusion length scale in the SP). How-ever, in our experiment the final SP radius is too largefor the current transfer to take place solely due to thismechanism. It appears that in general a current bounceoccurs due to a combination of the drop of the resistanceof the LDP and the impedance rise of the SP, which re-inforces the resistance drop in the LDP.

The fast current-escape from the SP found here mightbe related to the low current observed at the stagnationof Z-pinches [11, 20, 24–26], however further mechanismsfor this phenomenon of low current in the SP have to beinvestigated. The fast current redistribution might alsobear relevance to general pulsed-power systems, such asplasma switches [12–14] or transmission lines [15, 16]. Inthese systems, undesired plasmas of various properties

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can be produced due to different sources, such as elec-trodes [15, 16], instabilities [41, 42], evaporation in solidloads [43], boundary layers in gas loads [30], and unim-ploded plasma in Z-pinches [23]. Considering the variousmechanisms that can lead to a fast current redistribution,as discussed here and in [37] based on analytic modelsand simulations, the presence of such plasmas might thusaffect the operation of many laboratory plasmas. Fur-ther experimental investigations of the current switch-ing should help in developing methods to control it, andthus to achieve higher efficiency in current-pulse trans-mission and in producing higher energy-density plasmas[23]. Due to the fundamental aspects of this phenomenonit can also be relevant for the investigations of space plas-mas [44, 45], as well as the evolution of the current distri-bution and energy balance in fusion plasmas [5, 10, 20].

U. Shumlak gratefully acknowledges support of theErna and Jakob Michael Visiting Professorship at theWeizmann Institute of Science. This work was sup-ported in part by the Cornell Multi-University Centerfor High Energy Density Science (USA), by NSF-BSF(USA-Israel), by the Naval Research Laboratory (USA),by Sandia National Laboratories (USA), by the IsraelScience Foundation, by the German Science Foundation(DFG TR18), by NNSA 83228-10966 [Prime No. DOE(NNSA) DE-NA0003764], and by NSF PHY-1805316.

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