Theoreticalz-pinch scaling relations for thermonuclear-fusion experiments

We have developed wire-array z-pinch scaling relations for plasma-physics and inertial-confinement-fusion (ICF) experiments. The relations can be applied to the design of z-pinch accelerators for high-fusion-yield (~0.40.3em0exGJ∕shot) and inertial-fusion-energy (~30.3em0exGJ∕shot) research. We find that (δₐ∕δRT)∝(m∕)1∕4(RΓ)^-1∕2, where δₐ is the imploding-sheath thickness of a wire-ablation-dominated pinch, δRT is the sheath thickness of a Rayleigh-Taylor-dominated pinch, m is the total wire-array mass, is the axial length of the array, R is the initial array radius, and Γ is a dimensionless functional of the shape of the current pulse that drives the pinch implosion. When the product RΓ is held constant the sheath thickness is, at sufficiently large values of m∕, determined primarily by wire ablation. For an ablation-dominated pinch, we estimate that the peak radiated x-ray power Pᵣ∝(I∕τᵢ)3∕2RΦΓ, where I is the peak pinch current, τᵢ is the pinch implosion time, and Φ is a dimensionless functional of the current-pulse shape. This scaling relation is consistent with experiment when 130.3em0exMAI200.3em0exMA, 930.3em0exnsτᵢ1690.3em0exns, 100.3em0exmmR200.3em0exmm, 100.3em0exmm200.3em0exmm, and 2.00.3em0exmg∕cmm∕7.30.3em0exmg∕cm. Assuming an ablation-dominated pinch and that RΦΓ is held constant, we find that the x-ray-power efficiency ηₓ≡Pᵣ∕Pₐ of a coupled pinch-accelerator system is proportional to (τᵢPᵣ7∕9)^-1, where Pₐ is the peak accelerator power. The pinch current and accelerator power required to achieve a given value of Pᵣ are proportional to τᵢ, and the requisite accelerator energy Eₐ is proportional to τᵢ². These results suggest that the performance of an ablation-dominated pinch, and the efficiency of a coupled pinch-accelerator system, can be improved substantially by decreasing the implosion time τᵢ. For an accelerator coupled to a double-pinch-driven hohlraum that drives the implosion of an ICF fuel capsule, we find that the accelerator power and energy required to achieve high-yield fusion scale as τᵢ0.36 and τᵢ1.36, respectively. Thus the accelerator requirements decrease as the implosion time is decreased. However, the x-ray-power and thermonuclear-yield efficiencies of such a coupled system increase with τᵢ. We also find that increasing the anode-cathode gap of the pinch from 20.3em0exto0.3em0ex40.3em0exmm increases the requisite values of Pₐ and Eₐ by as much as a factor of 2.