An analytic model of radiative collapse of a Z-pinch

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Abstract
There is a critical current IPB of about 1 MA (the Pease-Braginskii current) at which Ohmic heating and Bremsstrahlung losses balance in a Z-pinch under pressure equilibrium. An analytic zero dimensional model shows the process of radiative collapse when the prescribed current exceeds the critical current. In particular for a linearly rising current radiative collapse is complete when the current is square root 3 IPB. However in practice the voltage limitation imposed by an external circuit prevents such a total collapse, and by including this in the model a maximum density ( approximately 1030-1032 m-3) can occur followed by an expansion and damped oscillation about an equilibrium at which the current equals the Pease-Braginskii current. In the absence of alpha-particle pressure the maximum density is limited by the resistance of the narrow column, the large voltage across which ( approximately 108 V) is balanced essentially by a large negative LI; it occurs when the current is IPB (( delta -1)/( delta -2))1/2 where delta =7/3+4/3 ln (Rw/a), where a is the pinch radius and Rw is the radius of the current return. The minimum current following maximum density is shown to be greater than IPB/ square root 2. Degeneracy effects can be included in the model.

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