A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review pinch models will be presented. A detailed account of the wire-array Z-pinch will be presented in section 5, with its application to hohlraums and inertial confinement fusion in section 6. In section 7 experimental results from other Z-pinch configurations will be discussed. Other applications of Z-pinches ranging from x-ray lithography to a magnetic lens for relativistic charged particle focusing will also be included in section 8. Earlier reviews of the Z-pinch include those by Kadomtsev [83], Haines [52, 84], Dangor [85], Peireira and Davis (especially on x-ray emission) [51], Matzen [86], Liberman et al [87] and Ryutov et al [88]. In addition there have been several volumes of papers from the International Conference on Dense Z-Pinch series [89] as well as special issues of the IEEE Transactions on Plasma Science [90], Laser and Particle Beams [91] and earlier Proceedings of Workshops on the Plasma Focus and Z-pinch [92]. 2. Physics of the equilibrium Z-pinch We first present some of the basic physical properties of a uniform Z-pinch in pressure balance. 2.1. Bennett relation This was first derived by Bennett in 1934 [26] for charged particles streaming with a uniform axial velocity. It can be extended to cover the general case of any current density distributions in a Z-pinch under a radial pressure balance which is described by ∂p ∂r =−J zB θ . (2.1) Employing the axial component of Ampère’s law, 1 ∂ r ∂r (rB θ) = µ 0 J z (2.2) we obtain, on integration B θ = µ ∫ r 0 J z r dr r 0 (2.3) and hence, from equation (2.1) ∫ ∂p ∂r =−µ J r z 0 J z r dr. r 0 (2.4) The ion line density N i is defined by ∫ a N i = 2πn i r dr, (2.5) 0 where a is the plasma radius. Taking the mean electron and ion temperatures to be T e and T i , respectively, we can write ∫ a N i k B (ZT e + T i ) = 2πpr dr, (2.6) 0 where Z is the ionic charge number and k B is Boltzmann’s constant. integrated by parts, as follows: ∫ a N i k B (ZT e + T i ) = π pd(r 2 ) 0 ∫ = [πpr 2 ] a r=0 − π r 2 ∂p ∂r dr ∫ a [∫ r ] = 0+πµ 0 J r r J z r dr dr, 0 0 Equation (2.6) can 11
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review where equation (2.4) has been employed. Writing x = ∫ r 0 2πJ zr dr this becomes Nk B (ZT e + T i ) = µ ∫ a 0 x dx = µ [ 0 x 2 4π 4π 2 0 or finally the Bennett relation 8πNk B (ZT e + T i ) = µ 0 I 2 (2.7) is obtained, where the total current I is defined by I = ∫ a 0 ] a r=0 2πJ z r dr. (2.8) Equation (2.7) is remarkable in that the average temperature of the equilibrium pinch can be calculated knowing only the line density and current. Another aspect of Bennett’s paper is the derivation of what is called the Bennett equilibrium. This is a special density (and current) profile associated with a spatially uniform axial velocity of current carriers and no ion flow. Taking this velocity as v ez the combination of Ampère’s law and pressure balance leads to − n e (r)e v ez = 1 ∂ µ 0 r ∂r (rB θ(r)) = 1 [( ∂ rkB (T e + Z −1 ) ] T i ) ∂ne (2.9) µ 0 r∂r n e (r)e v ez ∂r the solution of which is n eo n e (r) = (1+br 2 ) , (2.10) 2 where b = n eoµ 0 e 2 vez 2 8k B (T e + Z −1 T i ) . (2.11) Such an equilibrium is useful in the Hall fluid or the Vlasov models, but is not compatible with one based on collisional transport such as the parabolic profile which is based on uniform resistivity and axial electric field. There is an interesting feature of this profile; namely that the characteristic radius of the pinch, b −1/2 is given by b −1/2 c = λ D 2 √ ( 2 1+ T ) 1/2 i . (2.12) v ez ZT e Thus when the drift velocity approaches the speed of light c, the pinch radius shrinks to one Debye length, λ D = (ε o k B T e /n eo e 2 ) 1/2 , while a normal Z-pinch has v ez ≪ c and it has macroscopic dimensions much greater than λ D . The former could be relevant during a disruption (see section 3.14). This also shows the necessity of having a magnetically neutralizing return electron current in the fast ignitor where the fast electrons are indeed relativistic. 2.2. Particle orbits In the absence of ion centre-of-mass motion the ion radial pressure balance equation is 0 = Zn i eE r − ∂p i ∂r . (2.13) This equation shows that a radial electric field is required to confine the ion pressure. This is despite the fact that a magnetic field exists which can cause the ions to be magnetized, i.e. 12
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A review of the dense Z-pinch

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