^{1}and Stephen A. Slutz

^{1}

### Abstract

Presented is a semi-analytic model of magnetized liner inertial fusion (MagLIF). This model accounts for several key aspects of MagLIF, including: (1) preheat of the fuel (optionally via laser absorption); (2) pulsed-power-driven liner implosion; (3) liner compressibility with an analytic equation of state, artificial viscosity, internal magnetic pressure, and ohmic heating; (4) adiabatic compression and heating of the fuel; (5) radiative losses and fuel opacity; (6) magnetic flux compression with Nernst thermoelectric losses; (7) magnetized electron and ion thermal conduction losses; (8) end losses; (9) enhanced losses due to prescribed dopant concentrations and contaminant mix; (10) deuterium-deuterium and deuterium-tritium primary fusion reactions for arbitrary deuterium to tritium fuel ratios; and (11) magnetized α-particle fuel heating. We show that this simplified model, with its transparent and accessible physics, can be used to reproduce the general 1D behavior presented throughout the original MagLIF paper [S. A. Slutz et al., Phys. Plasmas 17, 056303 (2010)]. We also discuss some important physics insights gained as a result of developing this model, such as the dependence of radiative loss rates on the radial fraction of the fuel that is preheated.

We would like to thank D. B. Sinars and an anonymous referee for reviewing this manuscript. We also thank D. B. Sinars, M. E. Cuneo, M. C. Herrmann, C. W. Nakhleh, K. J. Peterson, and M. K. Matzen for supporting this project. We thank R. A. Vesey for useful discussions about approximations, MagLIF parameter scans, and for supplying the simulation input parameters needed for Fig. 4(k) in this manuscript. We thank K. R. Cochrane for supplying us with SESAME equation-of-state data and for suggesting the use of the Birch-Murnaghan formulation as we attempted to fit the SESAME cold-curve data to various analytic formulae. We thank S. B. Hansen for useful discussions about various simplified radiation loss models. We thank P. F. Schmit, P. F. Knapp, and M. R. Gomez for some helpful beta testing of the SAMM software. We thank M. R. Gomez for useful discussions about fully integrated MagLIF experiments. We thank M. Geissel for useful discussions about laser transmission through laser entrance windows. We thank the MagLIF/ICF and Dynamic Material Properties research groups, in general, for many useful technical discussions regarding various aspects of MagLIF; in particular, we thank A. B. Sefkow, M. R. Martin, C. A. Jennings, R. W. Lemke, T. J. Awe, D. C. Rovang, D. C. Lamppa, and W. A. Stygar. Finally, we thank the personnel of the Pulsed-Power Sciences Center (including the Z and ZBL facilities) at Sandia National Laboratories; without their hard work and dedication to excellence, this work would not have been possible.

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed-Martin company, for the United States Department of Energy's National Nuclear Security Administration, under Contract No. DE-AC04-94AL85000.

I. INTRODUCTION II. A SEMI-ANALYTIC MODEL OF MAGLIF A. Overview of the model and the radial distribution of the driving azimuthal magnetic field B. Circuit model for the Z pulsed-power driver C. Drive energetics and ohmic liner heating D. Liner dynamics and compression E. Liner equation of state, energetics, and ionization F. Fuel ionization, energetics, and adiabatic heating G. Fuel preheating (optionally via laser absorption) H. Fuel heating via magnetized

*α*-particle energy deposition I. Model of fuel hot spot and dense outer shelf regions J. Radiative losses K. Magnetized electron and ion thermal conduction losses L. End losses M. Magnetic flux loss due to the Nernst thermoelectric effect N. Erosion of the fuel's dense outer shelf region O. Primary fusion reaction rates P. Summary of the semi-analytic MagLIF model III. MODEL VERIFICATION IV. SUMMARY, FUTURE WORK, AND CONCLUSIONS

### Key Topics

- Magnetic fields
- 33.0
- Thermal conduction
- 28.0
- Ionization
- 16.0
- Laser heating
- 16.0
- Fusion fuels
- 15.0

##### G21B1/00

##### G21B1/03

##### H05H1/02

^{10}By contrast, the Z accelerator can also be put into pulse shaping mode to obtain a slower initial current rise, which results in a shockless (“quasi-isentropic”) compression of the liner wall throughout the duration of the liner implosion.

^{6–8,10}The magnetic field diffuses into the liner quite differently in these two cases. For the shocked case, the heating from the shockwave causes the liner to melt, which decreases the liner's conductivity, and in turn allows the magnetic field to diffuse into the liner rapidly, just behind the shock front.

^{10}For the shockless case, ohmic heating and melting eventually enable the field to diffuse in.

^{6–8}The diffusion rate is much slower for the shockless case than for the shocked case.

_{r}to a very small value.

_{Ω}would be zero for this case, even though ohmic dissipation is occurring). However, this steady-state ohmic dissipation is negligible relative to the dynamic dissipation discussed in the text because δ

_{skin}< r

_{l}− r

_{g}. For more on the partitioning of supplied electromagnetic energy into ohmic and magnetic channels when the associated electrical currents are time-dependent and spatially distributed, see Ref. 69.

_{g}(t

_{ph}), then only the hot spot region will exist.

_{h}/r

_{g}change relatively slowly from one time step to the next. However, a detailed description of our efforts in this regard is beyond the scope of this paper.

_{z}(r)/ρ

_{g}(r) = const., the normalized B

_{z}(r) would fall directly on top of the normalized ρ

_{g}(r) without this shift.

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### Abstract

Presented is a semi-analytic model of magnetized liner inertial fusion (MagLIF). This model accounts for several key aspects of MagLIF, including: (1) preheat of the fuel (optionally via laser absorption); (2) pulsed-power-driven liner implosion; (3) liner compressibility with an analytic equation of state, artificial viscosity, internal magnetic pressure, and ohmic heating; (4) adiabatic compression and heating of the fuel; (5) radiative losses and fuel opacity; (6) magnetic flux compression with Nernst thermoelectric losses; (7) magnetized electron and ion thermal conduction losses; (8) end losses; (9) enhanced losses due to prescribed dopant concentrations and contaminant mix; (10) deuterium-deuterium and deuterium-tritium primary fusion reactions for arbitrary deuterium to tritium fuel ratios; and (11) magnetized α-particle fuel heating. We show that this simplified model, with its transparent and accessible physics, can be used to reproduce the general 1D behavior presented throughout the original MagLIF paper [S. A. Slutz et al., Phys. Plasmas 17, 056303 (2010)]. We also discuss some important physics insights gained as a result of developing this model, such as the dependence of radiative loss rates on the radial fraction of the fuel that is preheated.

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