A two-layer single shell magnetized target for lessening the Nernst effect

Fuel magnetization significantly lowers the required radial convergence, enabling cylindrical implosions to become a promising approach for inertial confinement fusion. The Nernst effect on the two-layer single shell magnetized target design applied to a Z-pinch benefits from a gold layer that decreases fuel demagnetization and serves as a magnetothermal insulation layer, preventing magnetothermal losses. The resistive diffusion and Nernst advection of the magnetic field are considered in the radiation magnetohydrodynamic model, which alter the evolution of magnetic flux in the magnetized target and result in plasma demagnetization. The results demonstrate that targets with a wide range of parameters can achieve ignition conditions under a 30 MA driven current. A two-layer single shell magnetized target for lessening the Nernst effect has the potential to achieve ignition conditions. The fusion yield of the optimal target increases by 168% from 0.71 MJ to 1.90 MJ, compared to a one-layer single shell target.


Introduction
Highly compressed magnetic fields when applied to inertial confinement fusion (ICF) targets can increase the efficiency of coupling energy to targets and reduce the required energy of the driver [1][2][3][4].Previous experiments at the OMEGA laser facility used an 8 T magnetic field to improve the yield by 30% [5].Z-pinch magnetized linear inertial fusion (MagLIF) produced up to 1.1 × 10 13 thermonuclear deuterium-deuterium neutrons [6].Simulations predicted larger yield enhancements for targets with a higher applied field strength [7] and high-gain targets were envisaged to overcome the limitations in gain.A potential high-gain configuration for the MagLIF approach is designed with a dense cryogenic layer of deuterium-tritium (DT).Alpha particle slowing in the cryogenic fuel layer is predicted to produce about 1 GJ yield at a 60 MA drive current [8].The necessary condition for attaining magnetized target fusion ignition in uniform DT cylinders with axial magnetic fields is described as the product of BR being greater than 6 × 10 5 G cm −1 (B is the magnetic field strength and R is the fuel radius) and temperature close to 7 keV [9].
Multi-shell targets, such as double-shell [10] and revolver [11][12][13], have recently gained attention in ICF as a potential concept seeking to achieve ignition at lower ignition temperatures than single-shell concepts.Double-shell capsules were explored using a laser indirect drive and resulted in a robust burn metric at the national ignition facility [14].The volume ignition design of double-shell capsules was also considered in Z-pinch dynamic hohlraum (ZPDH) [15].Analytical models were proposed for a double-shell capsule driven by ZPDH and the fusion gain was expected to exceed 10 for a 50 MA peak current.Then, double-shell capsules driven by ZPDH were optimized [16].It is found that the fusion energy yields from one-dimensional (1D) simulations are quite stable in a quite large parameter space.
On the one hand, double-shell targets may be susceptible to fluid instabilities [17].The double-cylinder target was used to study hydrodynamic instability growth for ICF applications due to its convenience to diagnostic access while retaining convergence effects.In 2005, double-cylinder experiments were performed at OMEGA [18].The instability grows while a shock travels through the interface between substances of different densities.On the other hand, an initial density gradient target was studied to provide a possible solution to Rayleigh-Taylor instability growth [19].Investigations of the accuracy between experimental double-shell target implosion dynamics and 1D simulations, which inhibited the investigation of double-shell targets for a long time, can be traced back to 2000 [20].With the development of better target fabrication techniques and decreasing the instability effects, one target design achieved over 50% of the 1D calculated yield.
Alternative targets designed in recent years commonly have an inner layer of high-Z metal surrounding the fuel.High-Z materials have high albedos and can re-radiate most of the radiation coming from the hot fuel [13,21,22].This reduces the required ignition temperature and the convergence ratio (C vg = R f0 R fs ), as well as the instability growth.In 2019, Dewald et al added the high-Z layer in the pushered single shell target to study the physics of mix and radiation trapping [23].In 2020, Dodd et al investigated the time-dependent albedo of a high-Z wall in the revolver capsule to clarify the physics of radiation trapping [22].In the case with a high-Z layer, the losses of energy through radiation are restrained.When considering a high-Z layer in a magnetized target, the thermal loss from the hot compressed magnetized plasma to the cold linear was dominated by transverse heat conduction, which was inhibited by the axial magnetic field.The corresponding loss of magnetic flux is dominated by advection and the Nernst effect [24].The Nernst effect moved magnetic fields down electron temperature gradients [25][26][27] with expectations of plasma demagnetization [28,29].In 2020, the Nernst effect was indirectly inferred in the MagLIF experiment through the observation that, above a certain laser preheat energy, the neutron yield plateaus [6].In 2022, Walsh et al showed the first direct evidence of Nernst in magnetized implosions [30].Due to the complexity of multi shell targets, an intermediate platform, the two-layer single-shell magnetized target design will be discussed in this paper.A two-layer single shell magnetized target combines a low-Z pusher and a high-Z magnetothermal insulation layer in a single shell, offering ease of manufacture.
A two-layer single shell magnetized target with high-Z layer applied to Z-pinch has the potential to achieve ignition conditions at lower implosion velocity than a one-layer single shell target.Targets filled with non-cryogenic fuel, such as DT gas, offered a low-cost option for exploring the approach feasibility [31].Thus, a cylindrical target filled with DT gas driven by a Z-pinch is proposed in this paper.A two-layer single shell target with Nernst advection resulting in better implosion performance than a one-layer single shell target is shown by 1D simulation in section 2. Simulation models are introduced in section 3. The Nernst effect on a two-layer single shell target is discussed in section 4. The optimization of the target parameters is discussed in section 5. Conclusions are given in section 6.

Two-layer single shell magnetized target design
The two-layer single shell magnetized target is composed of three layers: a low-Z pusher layer (outer shell), a high-Z magnetothermal insulation layer (inner shell), and a fuel layer, as schematically shown in figure 1(a).The magnetothermal insulation [32,33] layer is used to emphasize that the high-Z layer modifies the temperature gradients.The modified temperature lessens the Nernst effect and the magnetic flux loss.Meanwhile, the magnetic field transported to the high-Z layer can suppress heat conduction loss in the radial direction and considerably enhance the temperature.
Fuel magnetization (15 T) and preheat (∼250 eV) before compression (75 ns) significantly lowers the required radial convergence, enabling cylindrical implosions to become an attractive path toward generating fusion conditions.The low-Z pusher layer is the dynamic layer and the material is determined by the maximum current of the drive source.In this paper, a peak current of 30 MA and a pusher of beryllium (Be) is chosen, which are referenced from the cases of Sandia National Laboratories [2].Meanwhile, the fuel is initially set as 2.7 mm radius (R f0 ), 10 mm axial lengths and a 3 mg cc −1 density.The corresponding linear radius can be obtained with an initial aspect ratio A R = R 0 /(R 0 − R f0 ) = 6 to ensure the linear being robust against MRT instabilities, where R 0 is the initial linear radius, and R 0 = 3.24 mm is obtained.The high-Z magnetothermal insulation layer of gold (Au) and d Au = 2 µm is chosen as our pre-design, the Be pusher layer can be calculated as d Be = R 0 − R f0 − d Au = 0.538 mm, and the optimum parameters are shown in section 5.
Maintaining the temperature inside the linear owing to the opacity of the high-Z metal is one of the reasons for the Au layer design in a two-layer single shell target.Another contribution factor is that the temperature profile along the radius altered by the Au layer has the potential to decrease the loss of  magnetic flux based on the Nernst effect.The Nernst effect on a magnetized target accounts for temperature gradient-driven transport and causes the magnetic field to be transported out of the plasma.The inclusion of the Nernst term decreases the fusion yield by about 70% in a one-layer single shell target [2].
The two-layer single shell design aims to reduce fuel demagnetization and increase fuel yield.
The pre-designed two-layer single shell magnetized target parameters are shown in table 1, together with their performances obtained by simulation as shown in table 2. For comparison, the performances of a one-layer single shell magnetized target are also shown in table 2. A two-layer single shell target with Nernst advection results in better implosion performance, and its fusion yield is 130% higher than a single target.The yield of a two-layer single shell target without including the Nernst effect is only 57% higher than the one-layer case.Thus, the superior performance of the high-Z two-layer single shell target can be attributed to the Nernst effect.
These 1D simulations are useful for quick orientation parameter space.Adding a high-Z layer against the fuel modifies the fuel temperature and lessens the fuel magnetic flux loss in the high-Z layer through the Nernst effect.Upon deceleration and stagnation, the high-Z layer has the danger of introducing high-Z mix into the fuel and increasing radiation loss.It is difficult to separate the effects of the magnetic Nernst advection from the bremsstrahlung radiation.The experiments show that the central fuel temperature energy loss can be offset by enhanced radiation trapping in the high-Z mix region, leaving the core performance largely unchanged when adding a high-Z layer [23].The temperature modification from the high-Z mix will further change the Nernst advection of the magnetic field.This may be mitigated by the addition of a low-Z antimix layer between the fuel and the high-Z layer.The fusion yield with a low-Z (CH) antimix layer in our simulation is 1.47 MJ.Compared to a two-layer single shell target (1.63 MJ), the antimix layer design did not significantly change the use of a high-Z layer for mitigation of the magnetic flux loss.

Simulation model
The Lagrangian 1D RMHD model is implemented using the MULTI-IFE code [34] and its magnetic field extension [12,35].In a magnetized target, a sufficiently large axial magnetic field plays an important role in inhibiting radial thermal conduction during stagnation.The charged particles perform Larmor orbits about the field lines instead of freely diffusing.
In a form suitable for Lagrangian discretization, the equations describing the evolution of the mass density ρ, fluid velocity ⃗ v, specific internal energy ϵ, and magnetic field ⃗ B, are where the integrals are carried out over volumes, surfaces, and curves moving with the fluid.
is the electric field in a frame moving with the fluid, P is the matter pressure tensor, and ⃗ ϕ is the thermal flux.The expressions for the magnetic and electric fields are further derived.The momentum interchange of electrons with ions ⃗ R is made up, in the linear approximation, of a frictional force ( , which arises by virtue of a gradient in the electron temperature [36].The coefficients ᾱ and β are tensorial quantities that depend on the magnitude and direction of the magnetic field. Neglecting electron inertia (m e ≃ 0), conservation of electric momentum equation is expressed as Expressing the density of current as ⃗ J = qn e (⃗ v −⃗ v e ), the generalized Ohm's law is obtained For the axial magnetized cylindrical target, the quantity of interest is the axial magnetic field ⃗ B z .The 1D cylindrical geometry for the axial magnetic field, ⃗ B z × ⃗ J θ is in the direction of the radius and the ∇ • Pe = ∂P ∂r ⃗ v r term can be neglected.Thus, the 1D Ohm's law can be written as The order of magnitude of the friction force ⃗ R u,θ is and the order of magnitude of the thermal force ⃗ R T,θ is where l c is the characteristic length of the problem.The coefficients (α and β) are rational functions of the electron Hall parameter x e = τ e ω e , where τ e is the electron collision time and ω e = qB/m e is the electron cyclotron frequency.
When the thermodynamic pressure is larger than the magnetic pressure, it is important to compare the thermal force ⃗ R T,θ with the friction force ⃗ R u,θ .In this case, Ohm's law simplifies to where αme neq 2 τe is the resistivity coefficient and β q is the Nernst coefficient derived from the thermal force.From the equations (3.4) and (3.10), it is shown that most magnetic field moves with the plasma, which is called frozen-in-flow.Resistive diffusion and Nernst advection of the axial magnetic field [37] exit due to the presence of magnetic field gradients and temperature gradients.The magnetic field in the axial direction is divided into a frozen term, Joule term, and Nernst term The Nernst advection causes magnetic flux losses from the fuel, which behaves as a wave.The Nernst thermomagnetic wave was firstly investigated by Gurevich and Gel'mont in 1965 [38], and its velocity is defined in radial direction as (3.12) Manifestation of the Nernst effect, which requires a magnetic field perpendicular to the temperature gradient [39], which is inevitably presented in the field-compression situation.The axial magnetic field creates a Nernst wave carrying the magnetic flux.It decompresses the magnetic field in the fuel as the wave moves outward against the temperature gradient.At peak compression, the compressed magnetized fuel can experience significant temperature loss and magnetic flux loss.The temperature loss is mainly caused by electron conduction and the bremsstrahlung radiation loss.To mitigate these losses, an alternative design class (two-layer) uses a high-Z layer to modify the temperature gradient through bremsstrahlung.By inspecting equation (3.12), it is recognized that the mitigation of a temperature gradient reduces the Nernst advection velocity.The magnetic flux loss in the fuel near the high-Z layer decreases owing to the reduced Nernst advection velocity.The magnetic flux retained in the fuel near the high-Z layer will further mitigate the electron conduction and reduce the ignition requirements.The high-Z layer design has the potential to improve the fusion yield.

Reference cases
The model from Lindemuth and Kirkpatrick showed that significant gain could be obtained in magnetized fuel targets when a magnetic field provided magnetothermal-insulation [32].To reach a large magnetization regime in the fuel, efficient magnetic flux compression is required.The magnetic field is mostly frozen into the plasma motion, and the rest has been lost from the fuel primarily due to resistive diffusion and Nernst advection.The temperature gradients at the hot spot boundary amplifying the Nernst advection results demagnetization in the hot spot.The gold layer in a two-layer single shell target has the potential to perform as an effective magnetothermal insulation layer due to its radiation characteristics and reduced fuel demagnetization.
The typical implosion and the current are visualized in figure 2, which is set as I(t) = 30 MA • sin 2 ( π 2 • t 100 ns ), using the geometry of figure 1.The current provides the radial driving force compressing the liner.Preheat laser (t preheat = 75 ns) and pre-set axial magnetic field (15 T) enable fusion conditions under a relaxed driver.A large number of cells are chosen for the metal layer (both gold and beryllium) and the trajectories lie very dense indicating its higher density than the DT fuel.The stagnation times (maximum compression and peak burn rates) for the two simulations (the two-layer single shell target and the one-layer single shell target) are plotted for comparison, which are 130.5 ns and 130.1 ns, respectively, and the two-layer single shell case has a lower implosion velocity.
The blue and red solid curves in figure 3(a) show the variation of Nernst advection velocity at the boundary of the whole fuel in the two-layer single shell target and the one-layer single shell target, respectively.Due to the laser preheating at 75 ns, the temperature gradient causes the Nernst advection velocity to increase rapidly.After stagnation, the fuel is heated by fusion and a second increase of the Nernst advection velocity appears in both cases.It should be noted that a larger Nernst advection velocity in the one-layer single shell target before stagnation means a larger magnetic flux loss in the fuel.Thus, the general demagnetization from the Nernst advection is reduced in a two-layer single shell target owing to the gold magnetothermal insulation layer, which corresponds to the magnetic flux evolution in figure 4 (the red curves).In figure 3(b), the variation of the Nernst advection velocity at the boundary of the central 90% fuel of the two-layer single shell target is significantly higher than the velocity of the one-layer single shell target after 118 ns.This results from a higher temperature gradient at the boundary of the central 90% fuel.This indicates a rapid temperature descending at the boundary of the central 90% fuel and a larger magnetic flux loss from the hot central region at the beginning of the thermal nuclear reaction (118 ns) in the two-layer single shell target.Combining the less magnetic flux loss in the whole fuel and more loss in the center of the two-layer single shell target, less magnetic flux is lost by the rest 10% fuel near the metal, which corresponds to the evolution in figure 4 (the blue curves).
The red curves in figure 4 represent the total magnetic flux of the whole fuel.As shown by the red curves, little magnetic flux is lost before 80 ns in both cases.It is then gradually diffused and transported outside the fuel.The black curves indicate that the magnetic field is totally frozen into the central 90% fuel before 80 ns.Considering that the fuel is heated by the preheat laser, a large temperature gradient is produced and Nernst advection is gradually increasing.The magnetic flux loss in the center fuel increases after preheating.The blue curves represent the magnetic flux in the rest 10% fuel near metal liner.The small rising of the blue curves indicates that the magnetic flux diffused and transported from the center (black curves) is absorbed by the rest 10% fuel.
In figure 4(a), at the beginning of the thermal nuclear reaction (118 ns), hot spot produces a large amount of energy leading to the temperature rising rapidly.In the two-layer single shell target, the black curve decreases at 118 ns and approximately 24% of the magnetic flux initially introduced in the center fuel is lost, as compared to about 20% in figure 4(b).This indicates that the two-layer single shell target has a higher center hot spot temperature and results in a steeper temperature profile at the boundary of the hot spot boundary.This predicted temperature profile will also be discussed in figure 5.In figure 4(a), the magnetic flux loss in the rest 10% fuel near metal layer (blue curve) decreases at 118 ns and approximately 16% is lost, as compared to about 23% in figure 4(b) at the onelayer single shell target.This lower magnetic flux loss corresponds to a higher metal temperature in the two-layer single shell target.This is because a higher metal temperature results in a moderate temperature gradient at the interface between the fuel and the metal layer.A moderate temperature gradient at the interface causes a lower Nernst advection velocity.Furthermore, the magnetic field near the interface between the fuel and metal layer reduces the thermal conduction losses from the fuel.The higher magnetic flux remaining as shown in the blue curve tends to upgrade the fuel temperature.Until stagnation, the whole fuel of the two-layer single shell target lost 40% of magnetic flux, while the one-layer single shell target lost 43%.
The Nernst advection is proportional to the temperature gradient, and the magnetic flux is transported against the temperature gradient to the region with lower temperature.Figure 5 shows the temperature profile at stagnation.The blue curve in figure 5 represents the two-layer single shell target, the red curve is the one-layer single shell target, and the dashed lines are the interface between fuel and metal layer, respectively.The temperature profiles in the fuel for these two cases are significant different.There is a relative flatter radial temperature profile in the center (0-0.075mm) of the two-layer   single shell target.This will tend to produce larger yields in the hot spot.At the edge of the hot spot, the temperature drops rapidly as the blue curve is shown between 0.075 mm and 0.095 mm.Between the fuel and the metal layer, a lower temperature gradient appears in the two-layer single shell target.This is mainly due to the high radiation opacity of the gold layer, which causes more radiation energy to be retained inside the fuel.This moderate temperature gradient results in a lower Nernst advection velocity and decreases the magnetic flux loss.Magnetic field in the fuel near the metal layer improves the fuel temperature and further moderates the temperature gradient.In contrast, the beryllium layer in the onelayer single shell target, as shown by the red curve, results in a steeper temperature gradient in the fuel at 0.11 mm and

Parameter studies
The optimum input parameters are analyzed in this section.The computed yields from a series of MULTI simulations using the geometry of figure 1(a) and varied axial fields are shown in figure 6.In MagLIF, the Nernst effect causes the axial magnetic field to be transported in the radial direction.The gold layer in the two-layer single shell target has shown its modifications of magnetic flux, temperature and yield through the Nernst effect.Yields from simulations with the gold layer (blue) and without (black) are plotted as a function of the initial applied B-field strength in figure 6.The difference between these yields is particularly notable for low field strengths.This is because the Nernst effect is inhibited when x e ≫ 1, as is the electron thermal transport.The optimum B-field is 15 T for our design.More fuel can be burned in the two-layer single shell target owing to the flatter radial temperature in the center if the initial field is not too large.
Figure 7 shows simulated yields as a function of initial preheat time.The optimum initial preheat time is about 76 ns with a simulated yield of 1.68 mJ.The initial preheat time can be varied from 73 ns to 76 ns with less than a factor of 96% yield penalty.The simulated yield drops to about 671 J for an initial preheat time of 83 ns.This greater sensitivity is due to fuel compressed by the metal linear.A larger yield can be obtained only with the fuel preheated before compression (83 ns).
It is interesting to see how robust the two-layer single shell magnetized targets behave with a range of gold layer thickness (Au thickness).The initial magnetic field (15 T) and preheat time (75 ns) are fixed as shown in the pre-design case (section 2).Simulated yields are plotted as a function of the Au thickness in figure 8.Note that for the zero Au  thickness case (the one-layer single shell target), the yield (0.71 MJ) is less than for the two-layer single shell targets.This is because the gold layer with its high opacity feature maintains the fuel temperature and further modifies the magnetic flux through the Nernst effect.The optimum Au thickness is about 1.4 µm and the simulated yield is upgraded from 0.71 MJ to 1.90 MJ with a 168% improvement.The lower simulated yield in the cases with higher Au thickness than 1.4 µm may result from the thicker Au layer requiring more driver power.For a given 30 MA drive current, higher gold layer thickness may result in worse compression as well as lower simulated yield.In general, the two-layer single shell targets show better performance than the one-layer single shell case.

Conclusions
In this paper, a two-layer single shell magnetized target driven by a Z-pinch is proposed.The optimal target is designed with 2.7 mm deuterium tritium gas, 1.4 µm gold and 0.5386 mm beryllium in that order.The initial axial magnetic field is 15 T and the initial laser preheat time is 75 ns.The magnetohydrodynamic evolution of the targets is simulated through the RMHD code.The importance of the Nernst effect is discussed in this simulation model.The Nernst effect in the two-layer single shell magnetized target accounts for temperature gradient-driven transport and causes the magnetic field to be transported out of the fuel.The gold layer in the two-layer single shell target improves the temperature gradient at the interface between the fuel and metal layer, and reduces the Nernst advection velocity.A higher magnetic flux remains near the interface owing to the gold layer in the twolayer single shell target.The magnetic field near the interface reduces the thermal conduction loss from the fuel.Thus, the fusion yield increases to 1.90 MJ with 168% improvement in the optimal two-layer single shell magnetized target as compared to about 0.71 MJ without the gold layer.
These 1D simulations are useful for quick orientation parameter space.But Rayleigh-Taylor instabilities and the high-Z mix of liner material into the fuel will develop in 2D and 3D simulations.Since adding a high-Z layer against the fuel modifies the fuel temperature and lessens the fuel magnetic flux loss in the high-Z layer through the Nernst effect, it is difficult to separate the effects of the magnetic Nernst advection from the bremsstrahlung radiation.Upon deceleration and stagnation, the high-Z layer has the danger of introducing high-Z mix into the fuel and increasing radiation loss.The temperature modification from the high-Z mix will further change the Nernst advection of the magnetic field.This may be mitigated by the addition of a low-Z antimix layer between the fuel and the high-Z layer.The antimix layer design did not significantly change the use of a high-Z layer for mitigation of the magnetic flux loss.Further, low-Z antimix layer research moving toward a more stable high-Z layered design is underway.A high-Z dopant magnetized target is also used in our further work to study the temperature modification of the high-Z mix region and further effects of the Nernst advection.
Work is ongoing to optimize the MagLIF target design.A cryogenic layer, a density gradient layer, an antimix layer and a high-Z dopant magnetized target may be valuable designs for lessening Nernst effect in the future.

Figure 1 .
Figure 1.Schematic of the two-layer single shell magnetized target (a) and the one-layer single shell magnetized target (b).

Figure 2 .
Figure 2. Implosion diagram and driven current of the two-layer single shell target (a) and the one-layer single shell target (b).

Figure 3 .
Figure 3. Nernst advection velocity at the boundary of the whole fuel (a) and the central 90% (b) in two cases (the two-layer single shell target and the one-layer single shell target).

Figure 4 .
Figure 4. Magnetic flux in the two-layer single shell magnetized target (a) and the one-layer single shell magnetized target (b).

Figure 5 .
Figure 5.Comparison of the temperature at stagnation obtained by the two-layer single shell magnetized target and the one-layer single shell magnetized target.

Figure 6 .
Figure 6.Comparison of simulated yields are plotted as a function of the initial B-field.

Figure 7 .
Figure 7. Simulated yields are plotted as a function of the initial preheat time.

Figure 8 .
Figure 8. Simulated yields are plotted as a function of the Au thickness.

Table 1 .
Parameters of the pre-designed two-layer single shell magnetized target.

Table 2 .
Performance of the fuel in two-layer single shell magnetized target and one-layer single shell magnetized target.