Effect of a dynamic axial magnetic field on a preconditioned single-wire Z-pinch

In this study, the effect and mechanism of a dynamic axial magnetic field on a preconditioned single-wire Z-pinch were investigated experimentally and theoretically. Optical diagnostic methods, including shadowgraphy, interferometry, Faraday rotation, and Thomson scattering, have been used to measure the parameters of magnetized plasmas. Compression of the azimuthal and axial magnetic fields was observed, and the suppression of the plasma instability was recorded and analyzed. The results showed that an external axial magnetic field could reduce the plasma instability and non-uniformity, but prolong the implosion time and weaken the compression ratio. In the implosion process with an axial magnetic field, the plasma rotated at a speed similar to that of imploding, which could be regarded as a stabilization method. A simplified model of the diffusion and compression processes of a dynamic axial magnetic field was developed to investigate the conditions for maximizing the amplitude of the axial magnetic field. Subsequently, the snowplow model was used to calculate the effect of axial magnetic fields on the implosion process and energy conversion.


Introduction
Z-pinch plasma is a phenomenon wherein cylindrical plasma implodes, stagnates, and emits intense X radiation with ultrahigh-power electrical pulses (ranging from 1 TW to 100 TW, lasting 50-1000 ns) [1,2]. This process generates extremely high temperatures, densities, pressures, and magnetic fields. Thus, it is widely utilized in material science, high-energydensity physics, and laboratory astrophysics under extreme conditions [3,4]. It is also a potential means to achieve Inertial * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Confinement Fusion (ICF), such as Magnetized Liner Inertial Fusion (Maglif) or dynamic hohlraum [5][6][7][8]. In the dynamics of a Z-pinch, plasma density, and magnetic field distributions are essential components to ensure symmetry and stability [9,10]. However, the development of magnetic Rayleigh-Taylor instability (MRTI) substantially degrades the implosion quality [11][12][13]. Therefore, mass density and magnetic field stabilization have been proposed and implemented in Z-pinch plasma.
The prepulse current is a crucial method for regulating the mass density of the Z-pinch plasma [14]. Typically, it is a small-amplitude current pulse lasting several hundred nanoseconds [15]. By vaporizing the solid load before the main pulse, the prepulse can prevent the Joule heating of the wire and eliminate the initial seed for the MRTI. Prior research has shown that wire arrays can benefit from this technique by improving axial uniformity, suppressing plasma ablation processes, and increasing x-ray intensity [16,17]. Moreover, the ability to adjust the prepulse is convenient for regulating the mass distribution and implosion process of wire arrays, which has promising potential for experiments using multiwire ∼MA generators [18,19].
Furthermore, utilizing an external axial magnetic field is an effective technique for suppressing plasma instability in Zpinch experiments [20,21]. The compressed axial and azimuthal magnetic field outside the plasma generates a magnetic shear, which can suppress MRTI [22]. Generally, two primary methods were used to apply an axial magnetic field. The first method involves using a prolonged electrical pulse and Helmholtz coil to generate a quasi-steady magnetic field [23][24][25]. This method requires a larger Helmholtz coil to further improve the magnetic-field strength and duration, which increases the load inductance and reduces the driving capability of the generator. The second method involves creating a required axial magnetic field by incorporating a spiral structure into the current return path. This approach offers the advantages of simplicity of construction, low cost, and ease of installation [26,27]. Under this condition, the magnetic field varies with the strength of the pulse current, and the mechanism of the magnetic field stabilization is more complex.
The axial magnetic field has a significant impact on the evolution of plasma instabilities, energy-coupling processes, and kinetic behavior. Two-dimensional self-illumination, xray backlight, and interferometric images were used to diagnose this process. The results of liner Z-pinches showed that the plasma radius increased during implosion and stagnation, and a new helical structure was created on the surface, which considerably alleviated MRTI [20]. Moreover, highresolution spectroscopic measurements were used to diagnose self-generated plasma rotation during the implosion of a gaspuff Z-pinch with a uniform axial magnetic field. The direction of azimuthal plasma motion depends on the direction of the applied axial magnetic field [24]. Although previous studies have primarily focused on the magnetic field stabilization of Z-pinch plasma, there has been a lack of simultaneous studies on mass density and magnetic field stabilization. In addition, the coupling process between the axial magnetic field and imploding plasma remains unclear when the external magnetic field is dynamic.
In previous experiments, the helical structure of the return column was similar to the solenoid shape, and the specific parameters were not easily adjustable. Regarding the current return posts, we used 3D printing technology to fabricate a helical return column with a freely adjustable structure, thus verifying the practicality of this technology in the introduction of an axial magnetic field. Moreover, we experimentally investigated magnetized plasma stabilization by combining a preconditioned Z-pinch with a dynamic axial magnetic field. Optical diagnostic methods, including shadowgraphy, interferometry, Faraday rotation, and Thomson scattering, were employed to diagnose the parameters of axially magnetized plasma. The compression processes of the azimuthal and axial magnetic fields were measured, and the suppression of plasma instability was recorded and analyzed.

Experimental setup
The experiment was conducted on the 'Qin-1' generator, which is capable of producing a 450 kA pulse of duration 400 ns. It features an independently triggered prepulse generator with a capacity of 10 kA and a duration of 16 ns [28,29]. The load was a silver wire with a 30 µm diameter. It was converted into a gaseous state by activating the prepulse current 600 ns before the main current [30]. The current was recorded using a Rogowski coil placed beneath the cathode. The voltage signal was captured via a resistive divider linked to the highvoltage cathode in proximity to the load.
The side-on and end-on optical diagnoses were constructed. Side-on laser diagnoses, including interferometry, Faraday rotation, and Thomson scattering, were used to determine the electron density, azimuthal magnetic field, electron temperature, and ion velocity in one shot, as shown in figure 1. The 1064 nm and 532 nm probe lasers with a duration of 8 ns were generated by a high-power pulsed Nd:YAG laser, which was recorded using a photodiode (DET10A, Thorlabs). The 1064 nm probe laser with an energy of 3 J was employed for Faraday rotation. The exploitation of the 1064 nm laser quadrupled the effective sensitivity of Faraday rotation compared with the 532 nm laser for λ 2 scaling [31]. A shadow channel and two Faraday channels were established using two non-polarizers and a reflector. Two polarizers with a high distinction ratio (10 6 :1) were used to analyze the Faraday channels to measure Faraday rotation [32]. The energy of the 532 nm probe laser was also 3 J and split into two parts using a beamsplitter with a splitting ratio of 8:92 (Thorlabs BP208). 8% of the 532 nm laser was employed for Mach-Zehnder interferometry to measure the electron density. The azimuthal magnetic field can be calculated using the areal electron density and Faraday rotation. The error would be caused by the fluctuation in the laser energy with an average value of 10% [33]. Faraday and interference images were captured by Charge-Coupled Devices (CCDs) with an exposure time of 1 ns, thus realizing a temporal resolution of 8 ns. Also, CCD cameras (Atik 383L+) had good linearity close to 1, which could ensure that the image brightness is proportional to the laser intensity. About 92% of the 532 nm laser was employed in Thomson scattering to measure the electron temperature and ion velocity. The laser was focused on the plasma region using a lens with a focal length of 750 mm. The specific focus position was calibrated using a needle tip in vacuum. Scattered light was collected using two fiber arrays in two directions. Each fiber array was composed of 16 fibers with a diameter of 250 µm and distance of 300 µm. An ICCD (DH334T) coupled with a spectrometer (Andor SR750) were used for spectral imaging with a exposure time of 5 ns. The object-to-image ratio was 2:1, which corresponds to a spatial resolution of 210 µm. After the probe laser was shot out of the vacuum cavity, it was collected by the beam dump.
End-on diagnoses, including interferometry and Faraday rotation, were used to determine the electron density and axial magnetic field, as shown in figure 2. The probe laser was  set to pass through the load along the axial direction using a beam splitter above the load. Subsequently, it was ejected from the vacuum chamber via an elliptical reflector located below the load. The anode and cathode of the load were two circular plates with fan-shaped holes that were reserved for end-on laser diagnosis. The metal wire was fixed at the center of the circular plates, where a fan-shaped hole was located. In the experiment, due to the axial diagnostic setup requiring a mirror to be placed beneath the load, it resulted in a height elevation of the load. Consequently, radial diagnostics cannot be performed in this configuration. Hence, in one shot, we were constrained to select either radial or axial diagnostics to measure the plasma properties. Three kinds of sideon diagnostics including interferometry, Faraday rotation, and Thomson scattering can be conducted simultaneously. Also, two kinds of end-on diagnostics including interferometry and Faraday rotation can be performed simultaneously.
The current return structure was set up in two distinct configurations: eight straight return posts parallel to the load axis or four helical return current posts made by 3D printing technology, as shown in figure 3. To determine the strength of the applied dynamic axial magnetic field, we used Comsol software to simulate this process. The load was replaced by a 5 mm diameter copper rod and the input current was the current waveform measured by the Rogowski coil. Meanwhile, we employed a terbium gallium garnet (TGG) crystal with a 10 mm diameter and 5 mm height to produce Faraday rotation when a short-circuit load was connected to the cathode and anode. The simulation results showed that the distribution of the axial magnetic field was inhomogeneous, generally increasing with increasing radius, and the inhomogeneity is higher in the region close to the reflux column. The experimental results are slightly larger than the simulation results, which may be because the short-circuit load introduces part of the axial magnetic field component during the experiment.

The effect of axial magnetic field
A series of experiments were conducted with and without an axial magnetic field to investigate the impact on the dynamic behavior of the Z-pinch plasma. The electrical waveforms of current, voltage, and radiation are presented in figure 4. The stagnation moment was determined using the rapid current decrease in the waveform caused by the formation of the microdiode after stagnation. Generally, when the prepulse current is applied to the single wire, the metal wire would be transformed into a core-coronal state and keep expanding. Then the main current is applied and the core-coronal structure would implode. Therefore, when the time interval between the main current and prepulse is larger, the initial radius of the corecoronal structure will be larger and the stagnation moment will be postponed. As shown in the current waveform of figure 4, the time interval between the main pulse and prepulse in shot #221127 was shorter than in shot #220345, indicating a smaller initial plasma radius when the main current was applied. However, the stagnation time was delayed from 204 ns to 245 ns, indicating a reduction in the implosion velocity with an axial magnetic field. This was accompanied by a slight decrease in the intensity of radiation peaks, which indicated a reduction in the maximum compression ratio. However, the overall radiation duration was also shortened, suggesting that the axial non-uniformity was mitigated, resulting in a more concentrated stagnation point at different axial positions. Also, Figure 4. Waveforms of current (gray), voltage (red), and radiation signal (blue) of a preconditioned single wire Z-pinch without the external magnetic field (shot #220345) and with the external magnetic field (shot #221127), respectively. The prepulse was applied to the load 600 ns before the main pulse. The green solid arrow and dashed arrow represent the stagnation moment with and without the axial magnetic field.
according to the integration of the radiation waveform, the overall radiation intensity increased by a factor of 1.7 after the application of an axial magnetic field.
In the plasma experiment, the load initially transformed into a gaseous state owing to the effect of the prepulse and subsequently displayed snowplow implosion upon application of the main pulse. During this process, mass accumulation and magnetic field compression occurred simultaneously at the boundaries of the plasma. Shadowgraphy images captured from different shots with and without an axial magnetic field are presented in figures 5(a)-(e) and (g)-(k). The change of non-uniformity and MRTI instability was evaluated. Without an axial magnetic field, axial nonuniformity was observed in the plasma column before 150 ns. This nonuniformity was quantified by the ratio of the maximum to minimum radii at different axial positions, which was caused by the initial radial difference of metal wire. As displayed in figure 5(m), the linear growth rate of observed non-uniformity was suppressed and decreased by three times after applying an axial magnetic field.
There was virtually no MRT instability in the early stage of implosion, regardless of the presence of an axial magnetic field. After 170 ns, MRT instability started to develop, which could also be effectively suppressed by the external axial magnetic field. We used the maximum and minimum intensity of MRTI instability to determine the error bars. But as shown in figures 5(c) and (i), after applying an axial magnetic field, plasma at most locations showed good stability and only a small amount of plasma showed obvious MRTI instability. There may be a small amount of plasma with a relatively large amplitude causing a long error bar. It is the reason why the magnitude is not-outside the reported error bars at some time points. And this inhomogeneity may be due to the non-uniform distribution of the axial magnetic field, which was added to the revised paper. But the average amplitude of MRTI can be reduced by more than 50% with the application of an axial magnetic field. In addition, we can also see the difference in the distribution of electrons from the interference images at stagnation, as shown in figures 5(f ) and (l). In the absence of an axial magnetic field, the surface of the high-density plasma exhibits numerous stripe shifts, indicating the presence of abundant free electrons. This could be regarded as a characteristic of instability and perturbation development. With an axial magnetic field, the phase shift in the surface plasma was nearly imperceptible, signifying the role of the axial magnetic field in surface homogenization and the suppression of instability.
A side-on Faraday rotation diagnosis was utilized to measure the current distribution during the initial stages of implosion to gain further insight into the generation and evolution of instability, as shown in figure 6. Relative to the shadow image, Faraday image 1 shows a darkening trend and Faraday image 2 shows a brightening trend which was caused by the opposite settings of the polarization analyzers in two Faraday channels. The electron density was calculated using Abel inversion with Fourier series [34]. Upon application of an axial magnetic field, a gradient of density was observed in the measurable region, indicating the presence of a density shell. The average error of the magnetic field was 10%, which was caused by the fluctuation in the laser energy [33]. The distributions of the total current within the radius during the implosion phase with and without an axial magnetic field at similar moments were calculated by equation (1) and depicted in figure 6(c) where I(r) is the total current within different radii, µ 0 is the permeability in a vacuum, r is the radius, and B(r) is the magnetic field at different radii. During the implosion process, the current layer was located outside the dense shell, and the implosion velocity of the current layer was larger than that of the dense shell. This resulted in a continuous increase in the current density during this process. With the axial magnetic field, the current and dense shells were located at a larger radius owing to the lower implosion velocity. Also, the width of mass accumulation was faster and resulted in a dense shell. In addition, the overall width of the current shell decreased, leading to an increase in the current density. This suggested that the axial magnetic field resulted in a reduction in the implosion velocity of the dense shell, which caused a narrower current shell and an increase in the current density at the same implosion time. After t = 160 ns, the highdensity plasma boundary exhibited a higher electron density gradient, resulting in no measurable areal electron density. This indicated that the radius of the current layer is small and the overall current distribution is similar to that without an axial magnetic field.
Furthermore, the velocity distributions of the imploding plasma were measured by Thomson scattering. The diagnostic results are displayed in figure 7. In the Thomson scattering process, the Doppler shift is commonly observed when light interacts with moving particles, the particle's velocity induces a change in the frequency of the scattered light, known as the Doppler shift. After collecting the scattered light, it can be regarded as the convolution of the ideal scattered light spectrum and the instrumental broadening. The instrument broadening was determined by measurements from a mercury lamp and is denoted as 0.02 nm. The theoretical scattered light spectrum was determined by parameters such as electron temperature, ion temperature, plasma velocity, electron-ion relative drift velocity, and mean ionization degree. The electron density was estimated from the results of interference diagnostics, while the relationship between the mean ionization degree and electron temperature is calculated using the FLYCHK database under non-local thermodynamic equilibrium conditions [35]. By utilizing the method of least squares, the scattered light spectrum can be fitted with a Voigt line shape resulting from the convolution of the instrument broadening and the theoretical spectrum. This process enables the determination of information such as velocity and electron temperature.
According to the fitting results, the radial and angular velocities and electron temperature were obtained, as presented in figure 7 and table 1. During the early stages of implosion, there was a substantial presence of electrons at the shell position, allowing for the measurement of radial and angular velocities. The axial magnetic field induced rotational behavior in the imploding plasma. As the shell continuously implodes, the implosion velocity slightly increases, and concurrently, the rotational velocity of the plasma shell also rises. The shell compresses the internal plasma, resulting in a collective implosion, albeit at a slower speed compared to the boundary plasma. This collision process aids in suppressing instability [36]. The lack of pronounced rotation in the internal plasma indicates that the majority of the current is distributed near the plasma surface. This azimuthal shear flow on the plasma surface is considered a crucial stabilizing mechanism [37].
For this rotation, the prevailing explanation currently involves the existence of a radial magnetic field, possibly generated by the divergence of the axial magnetic field or eddy currents in the electrodes, exerting an azimuthal force on the plasma with axial currents. Unfortunately, we are unable to directly measure the radial magnetic field. To investigate the presence of the radial component in the magnetic field, we removed the short-circuit load and recalculated the magnetic field distribution. The results indicated the existence of a radial component. As the radius increased, the radial component also increased. Interestingly, the direction of the radial magnetic field was opposite in different axial positions and different radii. But, the direction of the axially averaged value of the radial magnetic field was pointed from a small radius to a large radius, as shown in figure 8. In this scenario, the plasma experiences an azimuthal force, giving rise to the phenomenon of plasma rotation. This aligns with the rotational direction of the plasma that we have measured. Hence, this radial magnetic field may play an important role in the plasma rotation.
As shown in figure 7(d), the IAW spectral feature exhibits two peaks with different intensities. Typically, this discrepancy is ascribed to a disparity in the flow velocity between the electron and ion fluids, referred to as the relative electronion drift velocity, u d . The primary consequence of this relative drift velocity is the displacement of electron damping

Shot
Location  concerning ion damping. Consequently, the two IAWs experience distinct degrees of damping. The wave with lesser damping will engender more pronounced scattering [38,39].

The compression of axial magnetic field
During this process, mass accumulation and magnetic field compression occurred simultaneously at the plasma boundaries. To verify this process, we measured the axial magnetic field distribution using an axial Faraday rotation. The specific diagnostic results, which include a 2D distribution and averaged radial results of the magnetic field, are presented in figure 9. The axial magnetic field compression process at different radii was mainly concentrated on the surface of the implosion plasma. The amplitude of the magnetic field was in the range of 18-40 T, which indicated that the magnetic field effectively penetrated and was compressed in the plasma during the implosion. The compression of the magnetic field can be evaluated using the magnetic Reynolds number, which can be approximated as [40]: where L is the plasma thickness of the confining field, T e is the electron temperature (eV), λ is the Coulomb logarithm, and T impp is the implosion time of the volume. Based on the results of Faraday rotation and Thomson scattering during the implosion, it was observed that a toroidal plasma sheath with a thickness of approximately L = 0.1 cm can be created, while the electron temperature is estimated to be approximately T e = 20 eV. The Coulomb logarithm shows only a slight variation with the plasma conditions, and for these calculations, it was taken as 10 [41]. The implosion time, T impp , was estimated to be 200 ns, leading to R m = 5 for these experiments. As a result, the axial magnetic field was compressed by the plasma during the implosion, which also coincided with the measurement of the axial magnetic field. Moreover, the azimuthal magnetic field effectively compressed the axial magnetic field through the plasma, and the maximum strength of the magnetic field was limited by the magnitude of the azimuthal magnetic field. The maximum value of the axial magnetic field was found at r = 2.3 mm since most of the current was distributed in this area.

Theoretical models
A snowplow model considering a dynamic axial magnetic field was utilized to analyze the impact of axial magnetic fields on plasma dynamics and magnetic compression. In this model, fully conductive plasma is swept up by an infinitely thin, current-carrying plasma shell or magnetic piston [42]. In the presence of an applied axial field, the trajectory of the dense shell can be calculated as where r p is the plasma shell radius, t is time, m is the instantaneous linear mass of the imploding plasma shell, B θ is the azimuthal field, δB z is the difference in the axial field inside the plasma shell, and µ 0 is the vacuum permittivity. The measurement of the axial magnetic field indicated that it could diffuse into the plasma shell and then be compressed. The compression of the plasma implied the freezing of the magnetic field inside it. However, the diffusion and freeze-in processes cannot coexist in the plasma shell. In other words, if the plasma shell reaches the magnetic freezing state, the axial magnetic field can no longer diffuse into the plasma shell. Subsequently, the axial magnetic flux in the plasma shell will be compressed. This compression process occurred only at the position of the plasma shell and does not affect the distribution of the magnetic field inside it. Therefore, we established an ideal model to study this process, where time point t 0 after applying the main current was defined as the critical moment, assuming that the axial magnetic field could diffuse into the interior of the dense shell before t 0 and be ideally compressed after t 0 . Therefore, the initial amplitude and distribution of the axial magnetic field during the compression process were determined using the current measured by the Rogowski coil at t 0 . The magnetic compression resulting from the axial magnetic fields was calculated using the following formula [43]: where B z (t) is the axial magnetic field in the plasma shell and varies with time, H(t) is the total axial magnetic flux in the plasma shell at t, r(t) is the plasma radius at t, which varies with time and implosion radius, and w is the thickness of the plasma shell. Parameters w and t 0 are essential in this calculation. To investigate the influence of different w and t 0 , we calculated the axial magnetic field when the plasma radius was 2.5 mm, which was at the same position as the plasma shell in figure 9.
According to the snowplow model, the outer radius is 2.5 mm at t = 147 ns. Therefore, in equation (4), t was set as 147 ns, w was defined as several values in the range of 0.2-1 mm, r(t 0 ) was determined with the snowplow model and H(t) was determined by the current measured by the Rogowski coil. Meanwhile, the snowplow model was calculated from a radius of 16 mm to 2.5 mm. Because the axial magnetic field amplitude varies less in this region, we used an average value of 0.2 T/100 kA for the calculation. When t 0 was varied from 0 to 147 ns, the average axial magnetic field amplitudes with different shell thicknesses were calculated when the plasma was imploded to a diameter of 2.5 mm, as shown in figure 10(a). As t 0 changed, the amplitude of the axial magnetic field first increased and then decreased. This was also consistent with the previous model: as t 0 increased, the initial effective magnetic flux compression area decreases, and the initial amplitude increased. At the same time, the values of different shell thicknesses affect the maximum axial magnetic field amplitude but do not change the corresponding t 0 . Based on this model, the maximum axial magnetic field amplitude can be achieved when it starts to effectively compress the magnetic field at t = 58 ns by mass regulation. According to the magnetic field measurements obtained in our experiments, the effective thickness of the axial magnetic field compression was in the range of 0.2-0.3 mm, which was consistent with the measured distribution of the axial magnetic field.
Subsequently, the snowplow model was coupled with dynamic magnetic diffusion and compression. The initial plasma distribution had a radius of 16 mm, and its mass distribution was measured via interference experiments at two different angles [30]. Calculations were conducted with and without an axial magnetic field. When B z ̸ = 0, three values of t 0 = 30, 60, and 90 ns were considered. For a specific t 0 , the implosion was not affected by the axial magnetic field until after t 0 , at which point the axial magnetic field compression was considered to begin. The evolution of the implosion radius of the plasma with and without axial magnetic fields over time is shown in figure 10(b). Furthermore, the kinetic energy of the plasma, as well as the energies of the angular and axial magnetic fields, were determined using this model [44]. It was evident that the addition of axial magnetic fields caused a decrease in the kinetic and angular magnetic field energies of the plasma, which was associated with a reduction in the implosion velocity. However, the amplification of the axial magnetic fields was more pronounced, and at later stages, its energy may surpass that of the angular magnetic field. In addition, the effect of the axial magnetic field is most significant when t 0 is 60 ns, which is consistent with the calculation in figure 10(a).

Summary
This study explored the impact and mechanism of the axial magnetic field on a preconditioned Z-pinch plasma. A dynamic axial magnetic field was generated through the helical current posts, which produced a significant compression phenomenon during plasma implosion. Compared to traditional helical return columns, this design offers advantages in terms of ease of optimization and opens up broader prospects for application. The external axial magnetic field could greatly reduce the plasma instability while also lengthening the implosion time and reducing the compression. Moreover, the plasma rotates at a speed similar to that of implosion with an applied axial magnetic field, which can be regarded as the stabilization method.
Furthermore, the calculated magnetic Reynolds number demonstrated that the preconditioned plasma was an effective magnetic field confinement. By integrating the snowplow model, diffusion and compression models of the dynamic axial magnetic field were developed. The maximum axial magnetic field amplitude can be achieved by varying plasma parameters. Also, the calculated results showed that the addition of axial magnetic fields caused a decrease in the kinetic and angular magnetic field energies of the plasma. Therefore, the magnetic field compression process can be manipulated by adjusting the plasma and generator parameters.