A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
singular ion trajectory as discussed in section 2.2. For the 3.5 GeV anti-proton beam at CERN
the Z-pinch lens would have opposite polarity. An early attempt at employing a Z-pinch
as a magnetic lens for focusing an energetic ion beam was made at Brookhaven National
Laboratory [662]. Unfortunately it failed after several hours of operation. At CERN a careful
programme of materials testing and prototype development led to the use of graphite rather
than tungsten electrodes; helium rather than hydrogen for the filling gas; and low-porosity
alumina for the insulation tube [663]. Over 20 000 successful shots at 14 s intervals were
obtained, limited only by the time available on the accelerator. Erosion rates of the various
components were measured, and, with modifications, a further 20 000 shots were made. This
proved the capability of the Z-pinch as a magnetic lens for high energy beams [664]. The
specification of the Z-pinch magnetic lens at CERN is a 400 kA current in a plasma column
2 cm radius and 29 cm in length, L. This gives a peak magnetic field of 4 T and a focal length
just greater than the length of the pinch. Explicitly the focal length f is given by
f =
L
β sin β , (8.5)
where
( ) qµ0 J 1/2
z
β = L
, (8.6)
2p z
J z is the pinch current density and q and p z are the charge and relativistic momentum of a beam
particle. With a line density of 2 × 10 22 m −1 and a Bennett temperature of a few eV the regime
for stability is firmly in the resistive stabilization regime in figure 16. Indeed the magnetic
Reynolds’ number S, given by equation (3.4), is 2.7, and resistivity dominates, leading to
enhanced stability. With high resistivity the skin effect is diffused, bringing azimuthal magnetic
field towards the axis, so that a uniform gradient could exist by the time of maximum pinch
diameter after the first bounce [665]. The occurrence of the inverse skin effect [110, 271] also
led to a strengthening of the magnetic field gradient near the axis. The range of reproducible
focusing is also consistent with the above discussion on stability regimes. A few per cent
of nitrogen added to the helium gave a reduced pinch diameter, and hence a higher focusing
strength. This is important as stability for 4 µs or several Alfvén transit times is required.
8.5. Hall acceleration and multipole fields
Hall acceleration employing closed loops of Hall current was found by Haines [666] and
independently by Hess [667, 668] at NASA.
If a short coil surrounds a Z-pinch, illustrated in figure 91(a), so that the radial components
of the magnetic field B r penetrate the plasma, it is found that the plasma is accelerated away
from the anode and towards the cathode. The reason for this is that the axial current density
J z in the Z-pinch plasma interacts with B r to give an azimuthal Hall current J ϑ , given by
ηJ ϑ =− J zB r
n e e . (8.7)
In turn J ϑ interacts with B r to give an axial J × B force which accelerates the plasma. Using
equation (8.7)itgives
ρ dv z
dt
=−J ϑ B r = J zB 2 r
ηn e e
(8.8)
showing that the direction of acceleration is independent of the sign of B r . The equal and
opposite axial force is on the coil, which experiences an anti-parallel J ϑ in the plasma on the
cathode side and a parallel J ϑ on the anode side, the latter causing an attractive force, i.e. a
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