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For the very first time, it has been demonstrated that it is now possible to make Nuclear Fusion generate more energy than is required for its equipment to operate, but it is not entirely clear as to how the strategy can be applied to a large-scale facility. On the other hand, the Magnetic Confinement method that is used by the likes of ITER is still some way off achieving the threshold, because of the very high temperatures required to achieve the Fusion in the first place, but is possibly easier to 'scale up' than the Inertial Confinement method.

Could a possible system that does eventually prove practical have Magnetic Confinement as the main mechanism, but to get a 'head start' on the heat required to achieve the initial fusion, Laser Inertial Confinement is the trigger? Once this initial heat is generated, it is then transferred to the remainder of the ingredients that are waiting to be fused by Magnetic Confinement. At periodic intervals, new 'laser pulses' may be required. To what extent is this likely to reduce the initial Energy requirements?

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Assuming this part of your post was correct, "...it has been demonstrated that it is now possible to make Nuclear Fusion generate more energy than is required for its equipment to operate", then you would be asking some valid questions.

This is where the issue lies however, the NiF experiment produced more energy from fusion than was input into the fuel, but neglected the radiative losses to the pellet, the losses due to inefficiencies of the laser, and let's not forget the (at minimum) 50% loss of the energy used to generate the electricity to run the system in the first place.

If looking at the NiF system from just the wall plug then they pumped in at least 300MJ of energy to get less than 3MJ out, a Q of less than .01 - in other words, they have a long way to go; but this is progress none the less.

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Proposals for Magneto-inertial fusion do exist and research is ongoing. Lasers are good at heating and magnetic fields are good at confinement, so combining them may make it possible to pass the Lawson Criterion for net fusion energy production with shorter confinement times than needed for pure magnetic confinement fusion and lower temperatures than needed for pure inertial confinement fusion.

Laser triggers are not likely to be helpful for fusion machines such as tokamaks, where keeping the plasma under control for long periods, not ignition, is the major challenge. Instead, magnetic confinement methods such as Z-pinches can be used to enhance laser initiated inertial confinement fusion. For example, Magnetized Liner Inertial Fusion (MagLIF) is being studied at the Sandia National Labs Z machine, where

… a centimeter-scale cylindrical tube, or ‘liner’, is filled with fusion fuel (deuterium gas on Z experiments) and axially pre-magnetized to 10–20 T using Helmholtz-like coils. The fuel is then pre-heated to an average temperature of 100s of eV via inverse Bremsstrahlung absorption of 527 nm photons from a kilojoule-class laser. Finally, the liner is radially imploded over ∼100 ns via the Lorentz force to velocities of ∼70 km s$^{−1}$ using up to 20 MA of current from the Z machine.

Magnetized target fusion similarly uses magnetic fields to extend the confinement time of an inertial confinement plasma from nano seconds to microseconds.

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